Stark Effect in Rydberg States of Helium and Barium

CTW. Lahane

STARK EFFECT IN RYDBERG STATES OF HELIUM AND BARIUM STARK EFFECT IN RYDBERG STATES OF HELIUM AND BARIUM Free University Press is an imprint of: VU Boekhandel/Uitgeverij bv De Boelelaan 1105 1081 HV Amsterdam

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© C.T.W. Lahaije, Amsterdam, 1989 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanically, by photocopying, by recording, or otherwise, without prior written permission from the author. VRIJE UNIVERSITEIT TE AMSTERDAM

STARK EFFECT IN RYDBERG STATES OF HELIUM AND BARIUM

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Vrije Universiteit te Amsterdam, op gezag van de rector magnificus dr. C. Datema, hoogleraar aan de faculteit der letteren, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der natuurkunde en sterrenkunde op woensdag 18 januari 1989 te 15.30 uur in het hoofdgebouw van de universiteit, De Boelelaan 1105

door Christiaan Theodoras Wilhelmus Lahaije

geboren te

Eindhoven

Free University Press Amsterdam 1989 Promotor: prof.dr. W. Hogervorst Referent prof.dr. RP.H. Rettschnick En dan nog iets mijn zoon, een laatste waarschuwing. Er worden te veel boeken geschreven en van al dat lezen wordje alleen maar moe.

Prediker 12,12

Aan ....euh, dinges Voorwoord

Na bijna vijfjaar is mijn promotie-onderzoek afgerond. Het resultaat, vastgelegd in de vorm van een proefschrift, ligt nu voor u. Geboeid door het fenomeen laser ben ik na mijn af- studeren in 1984 naar Amsterdam gekomen om een promotie-onderzoek aan te vangen op het gebied van laserspectroscopie aan atomen. De afgelopen jaren heb ik benut om mijn kennis op het gebied van lasertechnieken en van de quantummechanica te vergroten. Zoals duidelijk zal zijn ben ik dan ook degene die het meest van deze periode geprofiteerd heeft. Het inspirerend enthousiasme en de goede sfeer binnen de vakgroep atoomfysica zijn mede bepalend geweest voor de succesvolle afronding van mijn promotie-onderzoek. Op deze plaats zou ik dan ook iedereen willen bedanken die heeft bijgedragen aan het tot stand komen van dit proefschrift. Wim Hogervorst, als mijn promotor en directe begeleider was jij degene die het onderzoek stuurde. Je bood mij echter de vrijheid om experimenten zelf gestalte te geven. Je enthousiasme voor het experiment, je aanstekelijk optimisme en onze wetenschappelijke discussies zijn steeds een stimulans geweest. Je informele manier van werken en ongedwongen persoonlijkheid maakten het mogelijk ten allen tijde voor raad en daad bij je aan te kloppen. Prof. Dr. RJ'Ji. Rettschnick wil ik bedanken voor zijn bereidheid als referent op te willen treden. Jacques Bouma, jij vormde samen met Wim Hogervorst de kern van de groep. Dank voor je inventieve bijdragen op het technisch vlak. Onze ontspannende conversaties en jouw relativerende visie op wetenschappers en hun ambities zou ik nooit hebben willen missen. Met Ben Post en Mohamed Zaki Ewiss heb ik de eerste jaren van mijn promotie- onderzoek prettig samengewerkt. Mijn collega's gedurende de gehele promotie-periode waren Wim Vossen en Erwin Bente. In het halfduister van de kelder bracht Wim Vossen mij de beginselen bij van het uitlijnen van een ringlaser; mijn dank hiervoor. Dankbaar heb ik gebruik gemaakt van de door Erwin geschreven besturings- en analyse-programma' s. Tevens wil ik jullie bedanken voor de vele stimulerende wetenschappelijke discussies. Er heeft zich inmiddels een nieuwe generatie promovendi aangediend. Deze groep bestaat momenteel uit Robert de Graajf, Tony van der Veldt, Ilse Aben, Pieternel Levelt en Ger- rit Kuik. Vanaf deze plaats wil ik jullie bedanken voor de (veelal nachtelijke) assistentie die jullie tijdens het experimenteren verleend hebben. Tony van der Veldt in het bijzonder ben ik dank verschuldigd voor zijn medewerking aan de analyses. Wim Ubachs, de nieuwe vaste medewerker in de groep, wil ik bedanken voor het aanleveren van een aantal ideeën voor stellingen. Van alle studenten die hebben bijgedragen aan mijn promotie-onderzoek wil ik speciaal Stephen Lorié en Frans van der Meijs bedanken voor de plezierige samenwerking. I thank Dr. K. Sakimoto of the Institute of Space and Astronautical Science in Tokyo for our fruitful discussions concerning the theory of the Stark effect. De heren van de computergroep en van de mechanische en electronische werkplaatsen wil ik bedanken voor hun ondersteuning. Tevens wil ik de heer Pomper bedanken voor het verzorgen van de figuren in dit proefschrift. Werklast en tijdsdruk kenmerken de laatste maanden van een promotie-periode. Het persklaar maken van een proefschrift voor een zekere deadline behoort tot één van de meest zenuwslopende bezigheden van een promotie-onderzoek. Tegen mijn wil in werden sociale contacten verwaarloosd. Ik wil dan ook iedereen in mijn naaste omgeving bedanken voor hun begrip, belangstelling, steun en stimulans. Tenslotte wil ik Anja bedanken voor alle steun en vertrouwen. Contents

Chapter 1 Introduction and summary

Chapter 2 Theory 9 1. Introduction 9 2. Structure of two electron atoms 9 3. Stark effect in hydrogen 12 4. Stark effect in complex atoms; calculation^ methods 14 4.1 Fcrtubational treatment 16 4.1.1 Spherical base 16 4.1.2 Parabolic base 16 4.1.3 General features 17 4.2 Diagonalisation of the energy matrix 18 4.3 MQDT of the Stark effect 19 4.3.1 Basic outline 19 4.3.2 Basic functions 20 4.3.3 Frame transformation 21 4.3.4 MQDT for quasi-bound Stark states 22 4.3.5 MQDT for autoionizing Stark states 23 4.4 Implementation 26

Chapter 3 Stark manifolds of barium Rydberg states 29 1. Introduction 31 2. Background 32 3. Experimental setup 33 3.1 Atomic beam production and detection 33 3.2 Lasers 33 3.3 Energy values and intensities 33 4. Results and discussion 34 4.1 Determination of quantum defects of higher /-levels 34 4.2 Energy positions of manifold levels 36 4.3 Relative intensity distribution over a manifold 38 5. Conclusions 39 Chapter 4 Electric field induced avoided level crossings in bound Rydberg states of barium 41 1. Introduction 43 2. Experimental setup and results 43 3. Discussion 45 3.1 Theory 45 3 3 3.2 Avoided crossing F2 <-> SX 46 3 x 3.3 Avoided crossing F2 <-> Pi 48 4. Conclusions • 52

Chapters Stark manifolds and electric field induced avoided level crossings in helium Rydberg states 55 1. Introduction 57 2. Theoretical background 58 3. Experimental setup 60 4. Results and discussion 62 4.1 "Isolated" manifolds 62 4.2 Avoided crossings 64 4.2.1 Avoided crossings of two levels of adjacent manifolds 64 4.2.2 Threefold avoided crossings 68 4.3 Remarks 70 5. Conclusions 71

Chapter 6 Electric field effects in weakly autoionizing 5dnf J=5 states of barium 73 1. Introduction 75 2. Theory 76 3. Experimental setup and results 79 3.1 Experimental setup 79 3.2 Experimental results 80 4. Calculational results and discussion 83 4.1 Comparison MQDT method with experimental results 83 4.1.1 Input for MQDT program 83 4.1.2 Comparison between experiment and calculated m=0 spectrum 84 4.1.3 Spectra calculated for I m I = 0,1 and 2 components 86 4.1.4 Avoided crossings between the 5d63d level and n=60 manifold components 87 4.2 MQDT method versus diagonalisation procedure 88 4.3 General remarks 89 5. Conclusions 90

Samenvatting 93

Chapter 3 of this thesis is a reproduction of an article which appeared in "Zeitschrift fur Physik D: Atoms, Molecules and Clusters": Z. Phys. D: At. Mol. Clust. 7 (1987) 37. Chapter 4 is submitted for publication to "Zeitschrift fur Physik D: Atoms, Molecules and Clusters". Chapter 5 is submitted for publication to "Physical Review A". Chapter 6 will be submitted for publication in a slightly revised form. The articles are reproduced with the kind permission of the publishers. Chapter 1

Introduction and summary

When a neutral atom is exposed to a static electric field F, its energy levels shift, split and broaden. The shift and splitting effect is usually associated with the name of Stark [1], although Lo Surdo [2] discovered it contemporaneously in 1913 (using a different method). The broadening effect was predicted by Oppenheimer [3] in 1928 to be a tunneling phenomenon and was first observed two years later by von Traubenberg et al. [4]. They measured the quenching of spectral lines of the Balmer series in hydrogen in an electric field. The physics of atoms in external fields is of fundamental interest and is related to the general problem of the properties of isolated atoms in different physical environments [5,6]. Interest for this subject can be found in various fields of physics such as plasma physics [7], astrophysics [8,9,10] and precision time and frequency metrology [11]. Many atomic quantities, such as level energies, fine structure splittings etc., are known to high accuracy and are sensitive probes for external influences. On the other hand, knowledge of the external effects can be used to influence atomic or material processes. As an electric field is easily produced in the laboratory and can be measured precisely, a study of its interaction with isolated atoms (in excited states) is straightforward. Furthermore, from a theoretical point of view the basic atom-field interaction can be evaluated in an essentially exact form. In the past it was only possible to study atoms in their ground state or in low-lying excited states. Under those circumstances external fields are weak compared to the internal atomic fields. It was technically impossible to create laboratory fields which were sufficiently strong to significantly perturb atoms in their normal states. For instance, the electric field binding the electron in the ground state of hydrogen is 5xlO9 V/cm. The development of tunable dye lasers drastically changed this situation [12]. With this narrow-band, tunable light- source the selective excitation of high lying, so-called Rydberg states, became possible. A Rydberg atom, which can be hundreds of Angstroms in size, is extremely sensitive to external perturbations. For example for the n=40 state in hydrogen, the Coulomb field of the nucleus can be overcome by an external field of only 125 V/cm. Thus instead of generating huge laboratory fields the strong-field regime of atomic physics may now be studied in easily controlled conditions using Rydberg states. Placing an atom in an external field (electric as well as magnetic) complex spectra are observed [13]. For weak fields these spectra can qualitatively be understood considering the electric field as a small perturbation breaking the local -2-

(spherical) symmetry [14,15]. However, an atom becomes a new physical (and chemical) entity when it is exposed to strong fields [16,17]. For example a strong electric field creates cigar-shaped atoms with large induced electric dipole moments or results in field-ionization. This changes well-known characteristics and requires a revision of the concept of atomic structure and dynamics which is based on the picture of electronic motion in a central field. / In the present work the effect of an electric field up to moderate field strengths on atoms with two valence electrons outside closed shells, in casu helium and barium, is studied. This study is part of a comprehensive research program of these so-called two-electron systems using high resolution laserspectroscopic techniques [14,15,18,19,20]. In the independent- particle model the electron is assumed to move in the averaged field of the nucleus and the other electrons [5,6], and each individual electron may be characterized with its principal quantum number nj and the orbital angular momentum quantum number I,. This set of quantum numbers designates the so-called configuration. An atom is said to be in a Rydberg state when one of its valence electrons is excited to an orbit with high value of n. A Rydberg series is formed by varying n. For example, the bound Rydberg series in helium or barium are lsn/ or 6sn/ respectively. In the absence of perturbers Rydberg series behave regularly and with increasing n level energies converge to the ionization limit. The level energy is proportional to (n — H/)~2 where the quantum defect JJ.; is a measure of the deviation of this level from the corresponding hydrogenic value (Ji/=O). With increasing value of / the quantum defect decreases steeply. Also Rydberg series converging to excited states of the ion are possible. In this case both valence electrons are excited. Low-lying members of these series, especially in the heavier elements, may perturb the regular behaviour of the bound Rydberg series as they sometimes have an energy lower than the ionization energy. Doubly excited states above the first ionization limit will eventually autoionize into an electron and residual ion. As the applied electric field in the low-field case can be considered as a perturbation on the atomic system an obvious approach of dealing with the problem is perturbation theory. Starting from the ground state (or metastable states) only Rydberg states with low /-value are within reach of a one- or two-step laser excitation. Commonly these Rydberg states have a pronounced quantum defect and are non-degenerate. In combination with the characteristics of the electric field operator (dipole operator) this implies that in general no first order energy contribution is present and no linear Stark effect will be observed. A second order energy contribution arises through field-induced mixing of opposite parity states. In principle the number of states which is involved in this quadratic Stark effect is infinite. Fortunately, the contributions of these states decrease rapidly with increasing energy separation and only a limited number of states, depending on the desired accuracy, has to be considered in practice. Using mostly known data on states with lower /-values the analysis of the quadratic Stark effect may yield information on states with higher /-values, not directly studied in the experiment. In this -3-

way the Stark effect was used as a tool to extract information on electronic states not accessible from the ground state by optical excitation [14,15]. The energy separation between opposite parity states decreases rapidly for higher /-values (quantum defect rapidly drops to zero). These levels may be considered to be nearly degenerate and a gradual change from quadratic to the linear Stark effect may be observed with increasing /-value. This implies that the (near) degeneracy is lifted by the field and a fanning out of Stark levels is observed, the so-called angular momentum manifold. This occurs in barium for />3 (Chapter 3) and in helium already for />1 (Chapter 5). A better way to calculate these manifolds instead of using perturbation theory is diagonalisation of the complete energy matrix [21]. The diagonal elements of these matrix are the zero-field energies. The off-diagonals contain the electric field contribution. There is no selection rule for n in an electric field. This implies that in principle an infinite number of opposite parity states as well as an integral over continuum states, has to be incorporated. However, with increasing energy separation between the states involved the value of the corresponding off-diagonal matrix element decreases rapidly. The desired accuracy then defines the number of n-values to be taken into account and thus the size of the matrix. Com- monly the integral over continuum states is omitted. This method works well for bound and narrow autoionizing states for not too high field strengths. With increasing field strengths more and more n-values have to be included. In practice the limitation is the maximum storage capacity of the computer used and the accuracy of the numerical method. At high field strengths the influence of the continuum, leading e.g. to broadening of lines through the process of tunneling, cannot be neglected any more. However, it is not feasible to incorporate these effects in the diagonalisation procedure. A more elegant theoretical procedure of calculating Rydberg states in an electric field is the frame transformation approach. The basic concept is that the configuration space can be separated into two regimes where different dynamics prevail. In the short range region the Rydberg electron penetrates the ionic core and the many-particle nature of the system must be considered. A long-range potential governs the interaction between the Rydberg electron and the residual ion core in the outer regime. The Schrodinger equation is solved separately in each region. The solutions are matched together by imposing the proper boundary conditions for a well-chosen core radius. In the field-free case the long-range term is dominated by the Coulomb potential and the theoretical framework describing the method is generally called Multichannel Quantum Defect Theory (MQDT) formulated in the R-matrix- [22] or quantum defect formalism [23]. An important feature of this point of view is that the physics concerning the highly excited Rydberg atom for the long-range region is described in terms of a hydrogen-like system. -4-

As an external electric field only influences the long-range potential starting point for the MQDT description is knowledge of the hydrogenic behaviour in a field. In contrast to the Zeeman effect, the Stark effect is solvable for hydrogen as the Schrbinger equation is still separable in parabolic coordinates when hydrogen is exposed to an electric field [5]. The MQDT version adapted for the Stark, effect [24,25] then connects the electron's motion in the outer region expressed in parabolic wavefunctions with the solution for the short-range region described in terms of standard (zero-field) MQDT. In Chapter 2 this approach of calculating Stark spectra is described together with other general methods of treating the interaction between an atom and an electric field such as perturbation theory. Details on the experimental setup used to perform the Stark effect studies presented in this thesis are given in the relevant chapters (Chapters 3-6). In Chapter 3 results of a first, exploratory investigation of the effect of an electric field on highly excited states with small quantum defects (n around 40) in barium are given. In the presence of a weak electric field the field-induced excitation of opposite parity states close to the level excited in the absence of the field may be observed. Extrapolation of their energy shifts as a function of field strength results in values of quantum defects for these not directly excited higher /-states (45/ £7). In the second part of this chapter the onset of an angular momentum manifold is observed and discussed. Although the agreement between experiment and calculations with the diagonalisation method is satisfactory, a problem assigning the different M components is encountered. As the manifold is observed via excitation of a state with high /-value (1=4) at least 5 different IMI components are observed, making the recorded spectra complicated to unravel. Special attention was given to the behaviour of the *Pi level crossing the manifold. Due to a series perturbation levels of this lP{ Rydberg series with high value of n have smaller quantum defects then the higher /-series [19]. This small but not negligible deviation from hydrogen should result in so-called avoided crossings between the 'Pj level and t1:-, manifold states with increasing field strength. However, the complicated structure of the observed spectra prevented the observation of these phenomena. In Chapter 4 examples of such avoided crossings for the M=0 sublevels between 3F2 3 3 1 and Si and for the M=0 and IMI=1 sublevels between F2 and P! for n around 60 are dis- 3 3 cussed. The F2 <-> Si avoided crossing could be reproduced by diagonalisation of an 8x8 energy matrix including the electric field contribution. The coupling dominantly occurred via 3 ! 3 the intermediate Po level. The crossing of Pi and F2 was first observed as some not understood interference-like phenomenon [18]. It turned out that these profiles in fact represented 3 i the integrated signal of individual stages of the avoided crossing F2 <—> P\ over a small field strength range. The minimum separation gap of this avoided crossing was smaller than 3 the linewidth of the F2 level. The broadening of this level was caused by small field inhomogeneities determining the integration range. Due to a lack of accurate information on all -5-

possible opposite parity states involved the diagonalisation method could not be applied. A fitting procedure was designed to unravel the actual avoided crossing from the observed spectra. Avoided level-crossings are a consequence of the non-hydrogenic character of multi- electron systems (quantum defects) and their observation presents a stringent test for the diagonalisation procedure. Beautifully resolved examples of such avoided-crossing phenomena are presented in Chapter 5 for the simplest and, from a calculational point of view, best-known two electron atom helium. Due to the development of an effective metastable beam source the lsnp Rydberg series of helium could be populated in a one-step laser excitation [20]. Emphasis in this experiment is on the avoided crossings between Stark levels of manifolds originating from different n (n around 40). Starting from a metastable S stats and under conditions where the polarisation of the laserlight was parallel to the direction of the electric field only M=0 Stark levels, both in the singlet and triplet series, were excited. Together with knowledge of the position (quantum defect) of all zero-field states [26] this facilitates both the analysis and the calculation. Special attention was given to twofold avoided crossings between the outermost Stark levels of two adjacent manifolds. Also a threefold avoided crossing between Stark levels originating from three different n-manifolds was recorded. The diagonalisation procedure was used to calculate these avoided crossings. To reproduce the minimum separation gap down to the experimental accuracy of a MHz it was necessary to include in the calculation for the twofold avoided crossing four and for the threefold case even five n values. Although the agreement between the experimental and calculated results for the moderate field strengths used (Fm(tt= 35 V/cm) is excellent, other imper- fections in the calculational procedure are clearly demonstrated: the number of n values to be taken into account in going to higher field strength rapidly increases and will eventually exceed the maximum storage capacity of the used computer.

Chapter 6 deals with an experiment in barium where the continuum is involved. Here the evolution of autoionizing Stark states exposed to an increasing field up to field strengths where two adjacent manifolds interact is studied. These manifolds originate from the zero- field excitation of the 5d3/2nf J=5 (n around 60) autoionizing states from the metastable 2 5d ' G4 state. As the configuration interaction between the 5d3/2nf Rydberg series and its 6sj/2eh continuum channel is weak the 5d3/2nf states slowly autoionize and show a narrow linewidth facilitating CW laserspectroscopy above the first ionization limit [27]. In the electric field broad resonances of the 5dns, 5dnp and 5dnd autoionizing series are mixed in. In the spectra, apart from manifold components consisting of narrow Stark states, broad and narrow resonances in between the two adjacent manifolds are observed. To reproduce these spectra both the diagonalisation method [21] and the MQDT version of the Stark effect [25,28,29] have been evaluated. The diagonalisation method provides energy positions and only yields -6-

satisfactory results when it is applied to states with narrow linewidths. In the experiments on 5dnf autoionizing states this is the case for the m 2 2 substates. In MQDT the influence of the continuum is well accounted for. Calculations based upon this method give a qualitative explanation of the observed spectra. For example the narrow structures in between two manifolds turned out to be manifestations of different stages of the avoided crossing between a 5dnd sublevel and manifold states. However, the quantitative results depend heavily on the zero-field MQDT parameters for the 5dns, 5dnp and 5dnd series. As these values are not known sufficiently accurate a detailed comparison between experiment and calculation was not feasible. Nevertheless, in principle the MQDT version adapted for the Stark effect offers the most advanced method for calculating Stark spectra for autoionizing states as well as for bound states in strong electric fields.

References 1. J. Stark: Sitz. Akad. Wiss. Berlin 47 (1913) 932 2. A. Lo Surdo: Atti R. Accad. Naz. Lincei 22 (1913) 664 3. J.R. Oppenheimer: Phys. Rev. 31 (1928) 66 4. H.R. von Traubenberg, R. Gebauer and G. Lewin: Naturwissenschaften 18 (1930) 417 5. H.A. Bethe and E.E. Salpelen Quantum mechanics of one- and two-electron atoms. Berlin Gouingen Heidelberg, Springer 1957 6. R.D. Cowan: The theory of atomic structure and spectra. Berkely Los Angeles London, University Press 1981 7. V.L. Jacobs: In: Atomic excitation and recombination in external fields. New York London Paris Mon- treux Tokyo, Gordon and Breach Science Publishers 198S, and references therein 8. A. Dalgarno: In: Rydberg states of atoms and molecules. Cambridge, Cambridge University Press 1983 9. J. Dubau: In: Atomic excitation and recombination in external fields. New York London Paris Montreux Tokyo, Gordon and Breach Science Publishers 1985, and references therein 10. W.D. Watson: In: Atomic excitation and recombination in external fields. New York London Paris Mon- treux Tokyo, Gordon and Breach Science Publishers 1985, and references therein 11. Laser spectroscopy VIII. W. Persson and S. Svanberg eds. Berlin, Springer 1987 12. T.W. Hansch: Appl. Opt 11 (1972) 895 13. M. Fauth, H. Walther and E. Werner Z. Phys. D: At. Mol. Clust 7 (1987) 293 14. K.A.H. van Leeuwen. Thesis Vrije Universiteit, Amsterdam 1984 15. M.A. Zaki Ewiss. Thesis Vrije Universiteit, Amsterdam 1985 16. D. Delande and J.C. Gay: Phys. Rev. Lett. 59 (1987) 1809 17. H. Rottke and K.H. Welge: Phys. Rev. A 33 (1986) 301 18. E.R. Eliel. Thesis Vrije Universiteit, Amsterdam 1982 19. B.H. Post Thesis Vrije Universiteit, Amsterdam 1935 -7-

20. W. Vassen. Thesis Vrije Universiteit, Amsterdam 1988 21. ML. Zimmerman, M.G. Littman, M.M. Kash and D. Kleppner: Phys. Rev. A 26 (1979) 2251 22. MJ. Seaton: Rep. Prog. Phys. 46 (1983) 167 23. U. Fano and A.R.P. Rau: Atomic collisions and spectra, Academic Press 1986 24. D.A.Harmin:Phys.Rev.A24(I981)2491 25. K. Sakimoto: J. Phys. B: At. Mol. Phys. 19 (1986) 3011 26. W.C.Martin: Phys. Rev. A 36 (1987) 3575 27. E.A J.M. Bente and W. Hogervorst Phys. Rev. A 36 (1987) 4081 28. K. Sakimoto: J. Phys. B: At Mol. Phys. 20 (1987) 807 29. K. Sakimoto: Private communication Chapter 2

Theory

1. Introduction An atom is in a Rydberg state when one of its valence electrons is in a highly excited orbit Its energy may be expressed by a hydrogenic formula:

Here I is the ionization limit of the atom, n the principal quantum number and u. the quantum defect ( n-u^v is the effective principal quantum number). R is a constant related to the fundamental Rydberg constant R_ according to

with m, and M the electron and nuclear mass respectively. This so called Rydberg formula (Eq.(l)) describes a series of levels as a function of n converging to the ionization limit I. It resembles the well-known hydrogen formula where \i=0. So the quantum defect accounts for the deviations of the energy of a Rydberg level from the hydrogenic value. It results from the screening of the nucleus by the inner electrons destroying the pure Coulomb field near the ionic core. The excited electron penetrates this core (the more so for small values of the orbital angular momentum 1) and then experiences a non-Coulomb interaction resulting in a non- zero quantum defect However, the excited electron moves far outside the residual ionic core for a large part of its orbit. Here it is only subject to a Coulombic field, giving rise to the regular spacing of successive Rydberg levels approximately according to n~2. The quantum defect reflects the short range electron-core interaction [1].

2. Structure of two electron atoms In this thesis work atoms with two electrons outside closed shells have been studied [1,2]. The atomic structure of atoms can be determined by solving the Schrodinger equation. For multi-electron atoms the independent particle model in the central field approximation is generally used to calculate the atomic structure. In this model each electron moves indepen- dently in the averaged field of the nucleus and all the other electrons. The energy of an electron then only depends on its principal quantum number n and orbital angular quantum number /. So for a two electron system the configuration is designated by (ni/^OVz)- A Rydberg series converging to the first ionization limit then corresponds with msn/ where n/ is -10- the Rydberg electron and ms the lowest possible orbit (m=l for helium and m=6 for barium). The ground state is ms2. Also Rydberg series converging to higher ionization limits exist. In helium these so-called doubly excited states are located far above the first ionization limit and have no influence on the bound levels. In barium, however, many doubly excited configurations lie below this first ionization limit and perturb the regular behaviour along the Rydberg series 6sn/. The theoretical framework to describe the interactions of doubly excited states belonging to Rydberg series converging to higher ionization limits with the regular Rydberg series is Multichannel Quantum Defect Theory (MQDT) [3,4]. In the case that no bound doubly excited states are present (alkali atoms and helium) a single channel Quantum Defect Theory (QDT) results in this framework. Doubly excited states above the first ionization limit will, depending on the coupling-strength with the continuum, eventually autoionize into an ion and electron.

The simple hydrogenic picture presented in the introduction is too restricted for most atoms. Other interactions (can) play an important role and more quantum numbers are required to characterize an atomic state. These interactions are: 1. Coulomb repulsion 2. Relativistic interactions (spin-orbit, spin-other-orbit and spin-spin) 3. Hyperfine interaction. The coupling between open shell electrons and the static nuclear moment results in hyperfine structure. It will only be present when the atom has a non zero nuclear spin. The isotopes of barium and helium discussed in this thesis both have zero nuclear spin and thus show no hyperfine structure. Considering the other two interactions the Coulomb repulsion is usually dominant for low / bound states. This electrostatic repulsion between two electrons (l/r^) results in a splitting of the msn/ configuration into two terms characterized by total spin S (S=0 for a singlet and S=l for a triplet term) and total angular momentum L (L=/ of the excited electron). Under the influence of relativistic interactions (electron spin) the S=l term then splits into three fine structure levels specified by the value of the total angular momentum J=L+S (LS-coupling case). For an msn/ configuration J can take the values 1-1,1 and /+1 and the spectroscopic notation of such a state isM+1Lj. Both electrostatic and magnetic interactions scale as n~3 [1]. For autoionizing states and high-/ bound states relativistic interactions usually dominate the repulsion, which leads to jj-coupling. Now the appropriate coupling scheme will be jj'J with j=l+s with s the electron spin angular momentum of the individual electron.

In an external field (electric or magnetic) the magnetic quantum number ML is important. It gives the projection of angular momentum on the field axis. Thus in a spherical base a state can be labelled Imsn/LSJM> or ln/n'/'jj'JM>. -11-

In this thesis work the influence of an external static electric field F on the atomic energy levels is studied. The Schrbdinger equation without the field (zero-field case) is assumed to be solved. In this case its Hamiltonian is represented by H°. The interaction between the atom and the external field will add to the Hamiltonian the term

HF=-PF=erF=erFcose=ezF (3) where P is the electric dipoie moment of the atom and F is directed along the z-axis. Now the total Hamiltonian reads:

H=H°+HF (4)

v(z)

z=O

Figure 1: A: Sketch of the Coulomb potential (Ve), Stark potential (VF) and the total potential (Vm) in the z-

plane. Indicated is the three dimensional saddle-point (x) and core radius /z/=r0 ( doited curve at T < To only applies to hydrogen). B: Domain of spherical and parabolic symmetries in two dimensions. - 12-

3. Stark effect in hydrogen The potential for the hydrogen atom in an electric field (in a.u.) is: V(r)=-l/r+Fz (5) In Figure 1A a cut of V(r) along the field axis is shown schematically. The zero-field ionization threshold at W=0 is the abscissa. In an electric field the electron is prevented from mov- ing toward z->+°° by the rising Stark potential. For z < 0 the ionization limit is lowered to

the top of a finite potential barrier W=WS < 0 at z=-r, = -Vl/F. This so-called "classical ionization limit" (in a.u.)

Ws=-2-fc (6a) with

Fs = l/(2nf (6b) actually is a saddle-point of the three-dimensional potential (Eq.(5)). The possibility that the electron may escape to z —»—<» by tunneling through this barrier implies that all Stark spectra are, in principle, continuous. However, Bethe & Salpeter [2] give a value for the field strength

of about 2/3 Fs for which tunneling becomes noticeable i.e. when line broadening via coupling with the continuum starts. The static potential (Eq.(5)) is axially, but not spherically symmetric unless F=0. Conse- quently / will no longer be a good quantum number. Still the hydrogenic Schrodinger equation

[-1/2V2-l/r+Fz]v=lV\|f (7) is exactly separable in parabolic coordinates (£,,T),) [2]. They are related to the spherical polar coordinates (r,6,<|>) by: or \ Ti=r(l-cos0) or i\=r-z (8)

Due to this axial symmetry, the parabolic quantum number m is equal to m/ in a spherical base. For Fr2*:! the potential (Eq.(5)) is more or less Coulombic,.i.e. V(r) is approximately 8 spherically symmetric for the region T<:TS). A typical value of F=50V/cm= 10" a.u. implies 5 rs = 10 a.u., so that in fact this region is quite large. Figure IB depicts the overlap of the spherical symmetry of the core and the region of parabolic symmetry. In the transition zone a wavefunction may be represented equally well in the parabolic as in the spherical base. So a transformation U relating these basis sets can be found (see section 4). -13-

The Ansatz (9) for the wavefunction of the H-atom yields the one-dimensional differential equations:

(10a) 2

_ m2-\ 1-3 Fn _W (10b)

The separation constant P appears as an effective Coulomb charge in the equation for %, with the remaining charge l-f$ in the other equation. No general solution of these separated parabolic equations in terms of known special functions exists. However, it is still appropriate to consider them exactly solvable in the sense that they can be solved to machine precision using standard numerical methods.

V(g)

Figure 2: Plots of the potentials Vfy and V(r\) ofEqs.(Wa) and (10b). Dashed lines are the pure Stark potentials 1I4F% and -H4Fr\. The potential of the electron in % or TJ motion of Eq.(10) is illustrated in Figure 2. The only difference between the two potentials is the sign of the Stark part: 1/4FJ; or -1/4FTI (dashed lines in Fig.2). This determines the qualitative difference in the behaviour of the f and g functions. For any energy W the % motion is confined to the interval %a<%<%b while the allowed region for X\ motion is Tla < TJ < T|6 and ,T| > i\c (see Fig.2), i.e. there is a potential barrier with a maximum at W=WS. Thus the % motion is bound while the electron can escape to T] -» oo (see arrows Fig.IB). Since the solutions of f and g cannot be obtained in closed form the WKB approximation [2,5] is used to calculate the eigenvalues and to inspect the behaviour of the wavefunctions in various regions. -14-

The separation constant P follows from the Bohr-Sommerfeld quantisation rule for motion [5]: characterize an atomic state. These interactions are:

where tii is the parabolic quantum number for the Z, motion. From Eq.(l 1) it follows that the separation constant {$ will have a discrete value for any value of the energy W: P = p(FWmm) (12) Considering the r\ motion the electron will eventually escape through the barrier. Before this occurs it will oscillate a few times in the well (quasi-bound). This results in a number of nodes in the wavefunction g represented by the quantum number n2- The total number of nodes is given by [2]:

Usually in the field this number is still associated with the principal quantum number. A Stark state can now redundantly be labelled with quantum numbers (n,n1(n2,m) or, introducing a new quantum number k=ni-n2, be represented as lnkm>. For fixed n and m, k ranges over the values [n-lml-1, n-lml-3,....,-n+lml+l]. A Stark state lnkm> can be constructed from linear combinations of the spherical state ln/m>. The transformation is given by [6,7]: (14a) where the coefficients A# are given by (n-l)/2 (n-l)/2 I (m+k)/2 (m-lt)/2 m (14b)

When viewing the Stark states as permanent dipoles P their energy shift -PF°° -Pz will then be proportional to k (measure of the projection of the charge distribution on the field axis). Under the influence of the electric field states with a large value of k will shift more than states having k around zero i.e. the states will fan out from the degenerate level at F=0 with increasing field strength. This results in the so-called angular momentum manifold.

4. Stark effect in complex atoms; calculational methods The treatment of the Stark effect in non-hydrogen atoms is more complicated than in hydrogen (section 3) as the zero-field potential is no longer Coulornbic near the ionic core. The influence of the actual potential is represented by quantum defects (sections 1 and 2) and the result is a shift of levels (Eq.(l)). Thus levels are no longer degenerate at zero-field and the appropriate wavefunctions are no longer diagonal in the parabolic base. Several methods -15-

exist for calculating Stark spectra, which will be briefly discussed next.

1. Perturbation theory This is of course the time-honoured method for calculating energy contributions. Basic assumption is that the contribution of the perturbation, i.e. the electric field, to the energy is smaller than those of the other interactions. It can be applied either in a spherical base [8 and references herein] or in a parabolic base [9 and references herein]. The most general applica- tion to ncn-hydrogen energy levels has been formulated in MQDT [10].

2. Numerical intef:ation The numerical integration of the hydrogen Eqs.(10) is straightforward. One obtains eigenvalues as well as, among others, tunneling amplitudes, ionization rates, photo absorption cross sections at arbitrary field strengths to arbitrary accuracy. A comprehensive theoretical study of hydrogen spectra in an electric field was performed by Luc-Koenig and Bachelier [11,12]. However, the influence of the core yielding non-hydrogenic behaviour complicates the calculation for more complex atoms. This method will not be discussed further.

3. Diagonalisation of the energy matrix The diagonalisation of the interaction matrix in a basis of zero-field states directly provides discrete eigenvalues. Also oscillator strengths can be generated. For non- hydrogen atoms the matrix is expressed in a spherical basis [13]. Unfortunately this method does not work for W > W5 as continuum interactions are not included [14].

4. Multichannel Quantum Defect Theory of the Stark effect The basic idea of this method is to consider separately a region (region 1) where the electron-ion interaction prevails and a region (region 2) where the electron-electric field interaction is dominant [5,15,16,17,18,19]. This is analogous to standard (Multichannel) Quantum Defect Theory ((M)QDT) [3,4] where the motion of the electron under influence of the long range potential is separated from its short range behaviour. In the present case the long range potential is influenced by the electric field and parabolic coordinates have to be used. The short range potential has spherical symmetry and is limited to a core with boundary surface r=ro (few a.u.). The appropriate transformation between region 1, where photo absorption can take place, and region 2, where the electron's motion is influenced by the electric field, then generates the Stark spectra. So the core determines a lower limit for r (r0) for which the hydrogenic representation holds (see Fig.l). -16-

4.1. Perturbational treatment

4.1.1. Spherical base The first order energy contribution of the field is obtained by diagonalisation of the perturbation submatrix belonging to a specific degenerate level. It is assumed that the electric field only influences the Rydberg electron. A general matrix element of Hp reads:

=8(rns,m's)S(s,s')eF)PnTrPnidrjYrm-lcoseYimidCl (15) From Eq.(15) it follows that the total parity of the angular momentum integral is (-1)'+/+1. This means that Eq.(15) only yields non-zero values whenever / and /' have different parity. However, for hydrogen atoms the commonly excited low /-valued states all have pronounced, different quantum defects, i.e. are not degenerate. Thus in general no first order or linear Stark effect is observed. Only for hydrogen all /-levels within one n multiplet are degenerate. In this case the electric field lifts the degeneracy in / yielding the angular momentum manifold (see section 3). In general, when the electric field contribution is small compared to the fine structure energy, only the second order or quadratic Stark effect will be observed. The matrix element has the general form:

2 ^ I 1

With some angular momentum algebra this yields for the energy shift of a level:

(17) wilh do and a^ the, n/j-dependent, so-called scalar and tensor polarizability respectively [8]. a scales with n7. This quadratic Stark effect causes a shift and splitting of the isolated levels according to Iml.

4.1.2. Parabolic base The energy of a hydrogen level through second order is [2]:

w= + 2«z 2 16 In hydrogen, even up to fields at which the atom ionizes, the consideration of only the first- order Stark effect is usually adequate. Although there is no explicit m dependence in the first- order term it enters in two ways. Using Eq.(13), it follows that even and odd m states are ordered alternating (k will be even or odd); they cannot be degenerate in a field. Furthermore, -17- when Iml grows the maximum possible values of nj and 112 decrease, resulting in a maximum Stark shift dropping with Iml. The extreme case will be of course the circular state with n=lml+l for which ni=ri2=k=0 and the first order Stark shift vanishes. For non-hydrogen atoms a perturbational treatment in the parabolic base is not meaningful as already at zero- field the energy matrix is in this case not diagonal.

4.1.3. General features Considering the quantum defect as a perturbation to the hydrogenic level its first order contribution to the energy is [i//n3. The half width of the Stark energy multiplet is in first order 3/2n2F. This implies that for 3/2n2F>]Vn3 states with orbital angular momentum I can be assumed to be degenerate with all higher /-levels and a transition from quadratic to linear Stark effect can be observed. The quantum defect roughly scales with l~5 [20] so for higher values of / a relatively weaker electric field is required to generate angular momentum manifolds. Within one spectrum the linear (in already degenerate levels) as well as the quadratic (in still isolated levels) Stark effect may be observed (see Fig.4, Chapter 3). Some properties of Rydberg atoms are collected in Table 1. Included are rough esti- mates of the general features of manifolds obtained with the perturbational treatment. As a significant part of the work discussed in this thesis concerns bound states around n=40 the related hydrogenic values are also given in Table 1.

table 1: Properties of Rydberg states

Property n dependence (a.u.) n=40 Binding energy W n"2 68.6 cm-1 Spacing AW between adjacent n states n-3 3.3 cm'1 orbital radius n2 2350 a» dipole moment n2 radiative lifetime x n3 >60us polarizability a n7 classical ionization threshold F, n4 125 V/cm first crossing between two manifolds F n-5 = 16.5V/cm density of Stark state n4 (m=0) = 12 /cm"1 n3 (m=n-l) =0.3 /cm"1 number of crossings before ionization n2 (m=0) n (m=n-l) -18-

4.2. Diagonalisation of the energy matrix Although the perturbative treatment resulting in a power-series expansions in the field strength (section 4.1.) can give satisfactory results, it is in principle inadequate. For example the perturb?tive expansion is non-convergent [9] and only at low fields a good estimate of the energy eigenvalue can be made. Furthermore such a treatment is hardly suited for the description of an avoided crossing between levels of different n-manifolds [13]. V/hen all levels of a manifold are of importance a better approach to derive eigenvalues (and eigenvectors) is to diagonalise the complete energy matrix. The diagonal of this matrix contains the zero-field energies (Eq.(l)) and the off-diagonal elements contain the matrix elements of the Stark operator (Eq.(3)). In principle all values of n have to be included but fortunately the off- diagonal elements involved decrease rapidly in value as the energy difference between different n-states increases. The desired accuracy then specifies the number of n-values to be incorporated in the calculation.

10 15 20 25 -> F(V/cm)

Figure 3: Example of a calculated Stark map for helium (m=0). Angular momentum manifolds originating from n-39, 40 and 41 are shown. Energy positions as a function of field strength are given relative to the n=40 hydrogenic level. -19-

At first glance the zero-field parabolic representation provides an attractive basis for the Stark problem, as the non-hydrogenic Stark spectra resemble the hydrogenic manifolds. How- ever, this similarity presents practical difficulties: The zero-field energies for non-hydrogen atoms are eigenvalues (diagonal) in a spherical, not a parabolic, basis. So a transformation from the parabolic to the spherical representation is necessary anyway. Furthermore, because of the deviation from hydrogen (quantum defects) the Stark effect for non-hydrogen atoms is not diagonal in the parabolic basis. For these reasons the spherical representation has been chosen in the diagonalisation procedure. However, Stark states of a manifold will be labelled with parabolic quantum numbers to facilitate their identification. The matrix elements of the Stark interaction (neglecting angular momentum coupling) have the general form (compare relation (15)):

=8m',m8r±1jF (19) The angular part of the matrix elements can be evaluated using well-known Racah algebra. The radial matrix elements are calculated by numerical integration of the radial equation in the Coulomb approximation [21,13], The elaborated off-diagonal matrix elements for helium and barium, including the coupling of angular momenta, are given in Chapter 3 and 5 respectively. The resulting eigenvectors provide a measure for the relative intensities of Stark states. Stark maps are constructed connecting the calculated energy eigenvalues after diagonalisation per field strength for a series of field strengths. An example is given in Figure 3 for singlet helium, m=0 and n around 40.

4.3. MQDT of the Stark effect

4.3.1. Basic outline The most distinguished non-perturbative theory on the Stark effect in Rydberg spectra of bound and autoionizing series is MQDT adapted for the electric field case [5,15,16,17,18,19]. The approach of Harmin [15,16] follows the conventional line of QDT for a one-electron system but departs from conventional accounts of the Stark effect especially that of perturbation theory. Sakimoto [18] extended Harmin's theory and adapted it to multi-channel problems i.e. for two or more electron systems. Starting point of the quantum defect treatment is the distinction between short- and long-range effects. The short range effects are limited to the core (r < ro). Processes in this region are governed by the non-Coulombic ion-electron interaction and independent of the external electric field, as the electrostatic forces (1 a.u.) are much stronger than typical laboratory field strengths (< 10~6 a.u.) over atomic dimensions. This region is then adequately -20- represented by zero-field parameters of the atom, namely quantum defects and dipole matrix elements (photo absorption process). These quantities can be deduced from experimental data, as in this thesis, or fronvab initio calculations. In standard MQDT ([3], no electric field) the long range effects are determined by the Coulomb potential. Solutions of the Schrodinger equation in this region are the regular and irregular Coulomb functions given in the spherical base. MQDT then provides the electron's motion by the appropriate connection of the short and long range regions. This is represented by the so-called reactance matrix R° (upper index 0 referring to zero-field). For autoionizing states it is more common to consider the system as a scattering problem. In- and outgoing waves are formed as linear combinations of the regular and irregular Coulomb functions and a scattering matrix S° can be derived from R°. In principle the R° or S° matrix contains all information to calculate bound or autoionizing spectra respectively for the zero-field case. Applying an electric field only the long range term is affected. The Coulomb + Stark potential is given in Eq.(5). Parameters in this region are obtained from a WKB analysis of the Stark effect in hydrogen [5]. The core potential, though complicated, is spherically symmetric whereas the Schrodinger equation with potential (5) is separable only in parabolic coordinates (£,TJ,<(>). Both areas (r < ro and r > r0) can be linked together (boundary conditions) by realizing that for r >TQ there is a large region where the Stark potential in Eq.(5) is negligible compared to the Coulomb term (r

Coulomb region (20) thus marks the overlap of regions of spherical symmetry, r ro and the appropriate transformation between the two different solutions of the Schrbdinger equation in this common region connects the two distinct types of input parameters.

4.3.2. Basic functions The motion of the excited electron in an electric field is determined by potential (5). Functions f and g are the solutions of the two equations given by the separation of the Schrodinger equation with potential (5) in parabolic coordinates (Eq.(10)). Solutions f and g are obtained in a WKB approximation [5,18]. Both functions f and g are regular at %=0 and Ti=0 respectively. (M)QDT treatment also requires the use of the irregular solution (not defined at r=0) for the Schrodinger equation. As the electron can only escape to r\ —><» (see Figs.l and 2) only the behaviour of the function g is of interest in this case. The generalized WKB solutions of Eq.(lOb) are: -21-

14 g (Tl) = cCn) ((Jfc/2)*sinA Wx(p) + (jfc/2) cosA W2(p)] (21a)

14 li J(Tl)=c(Ti) [(fc/2) cosAW1(p)-(Jt/2) sinAVV2(p)j (21b)

Here g is the regular and g the irregular solution at T|=0. The precise form of the quantities c and k is not of special interest. The phase integral A is defined as follows [19]:

p is a variable related to t\ and Wi(p), W2(p) are the parabolic cylinder functions [18,19]. For

r\b < t] < r\c, Wi and W2 are exponentially decreasing and increasing functions with i\, respectively. Two independent solutions of the Schrodinger equation for r > ro are then: vim* (23a)

When the electric field contribution in potential (S) is negligible compared to the Coulomb term (r«rj) the spherical base is also a good representation. Two independent solutions of the Schrodinger equation in this region can now be given in terms of the regular (g) and irregular (g) Coulomb functions:

)=8wi(r) Ffc, W) (24a) (24b) where Y^ are spherical harmonics.

4.3.3. Frame transformation At small 4 and r\ an analytical expression for the regular wavefunction can be given [2] y(FW$m)=N(FW$m)(&T\)m/2(2K)-meim* (25) Here N is a normalization constant [18].

At r

(27) -22-

Ap; reduces to AH (Eq.(14b)) for the special case where the quantum defect ^=0 (v=n, see Eq.(l)). For v=n and arbitrary p, Fano [22] showed that Apj may be expressed as an analytical continuation of the 3j-symbols with real arguments. The frame transformation matrix Up/(FW>n) is defined by: p (28a)

Combining Eqs.(26),(27) and (28a) with the proper expressions for N and N° results in [18]: ji (29) where C is a normalization constant [11].

The frame transformation can be used over the (wide) region r0

4.3.4. MQDT for quasi-bound Stark states In this section an analytical formula for calculating quasi-bound Stark spectra is derived following the Multichannel Quantum Defect approach. Expressions are obtained according to the R-matrix formalism [3]. Spin variables are omitted for reasons of simplicity. The solution

of the Schrodinger equation for the system with total energy E < 0 for r :»r0 can be written as:

*(YP"O= Z Y7G(rP'/n';7pm) (30)

where Yy denotes the wavefunction of the core in its state y and G the solution of the electronic motion in potential (5). So G may be expressed as a linear combination of \i(FW$m) and \j/(/TVP/n) with W=E-Ey, (Ey being the energy of the core with respect to the ground state). In matrix notation [18]

r>r0 (31) The smooth connection of G to an inner (r < ro) solution at the core boundary determines the matrix R. For r < r^ a solution for the present system can also be expressed in terms of the spherical Coulomb functions (Eq.(24)):

R = UR°U' (33) The functions G represent in fact base functions for the electron motion. Boundary conditions have to be imposed on the actual electronic motion (H), which in general is given by a linear combination of G: (34)

where i (and j) abbreviates y,p and m, and L is a constant column vector. The condition of a

quasi-bound state is imposed by requiring that at t\b < T| < T|c (see Fig. 1) terms in H growing with T| are eliminated. So terms proportional to Wj(p) must vanish (see Eq.(21)). This yields:

W2(Pi) (cosA,-5y--Rij]Lj=O (35)

which is satisfied provided

0 (36)

Condition (36) is the extension of MQDT in the electric field case. When Rv=0 Eq.(36) gives the WKB approximation of the Stark effect in hydrogen. For F=0 Eq.(36) reduces to zero- field MQDT in the parabolic basis. With transformation matrix U (Eq.(29)) condition (36) can also be expressed in the common MQDT in spherical base.

det|tan(nvy)8y+/?j}|=0 (37)

where i (and j) abbreviates y,l and m. For one-electron systems (and for helium), which have no bound doubly excited states, the R-matrix reduces to (QDT):

fly=tan(jcn,)8y (38) where n is the quantum defect. Inserting Eq.(38) into Eq.(36) with the help of Eq.(33) yields:

det cotAp Sp-p - £Up'/ tanfaHf) Ufp =0 (39)

From this relation the Stark energies of quasi-bound states of alkali metals and helium may be calculated.

4.3.5. MQDT for autoionizing Stark states Again the total energy E of the system E=Ey+W (Ey being the energy of the ion) is preserved as a constant, but now both cases W < W5 and W > Ws arc possible, corresponding to closed and open (continuum) channels respectively. For the free electron (W > W^) the Stark effect can be neglected and its motion may well be described in terms expressed in the -24-

spherical base (Eq.(24)). The Staik effect of the Rydberg electron in a resonance state

(W < W5) is fully taken into account To discuss the autoionization process (as well as the reverse process dielectronic recombination) it is more convenient to employ the S-matrix formulation [3,19]. Introducing the functions (40a) (40b) the electronic part of the solution of the Schrodinger equation (Eq.(31)) can be rewritten as:

G=y.-Vf+x (41) As for W > Wj the Stark effect is neglected, the functions y+ and \|/2 represent outgoing and incoming waves, respectively, for open channels. From Eqs.(31), (40) and (41) % is related to Rby X = (l+iR)(l-«r1 (42) Since U is not orthogonal, % cannot be direcfly related to its zero-field quantity by a frame transformation such as Eq.(33). However, % is unitary and symmetric [19]. When a doubly excited state above the first ionization limit of an atom is occupied it will autoionize (producing an electron and a residual core) due to coupling with the continuum. Depending on the strength of this coupling the autoionizing state will have a certain intrinsic linewidth. Therefore not only resonance positions but also cross sections should be calculated. For bound (a) -• free (j) transitions the oscillator strength is given by [1]:

2 <»F,(E)ldl>Fa>| (43) where ^jiE) and *Fa are normalized according to:

=Sjr 5(£-F) (44a) / <^ = 8afl' (44b) The wavefunction *?;(£) describes the actual motion of the ionized electron and the residual ion core. This wavefunction ¥)(/?) can be expanded in terms of eigenfunctions of the ion core:

y (45) yaw where this eigenfunction of the core Yy is normalized to

=5Yy' (46) If Ey>E (i.e. W'<0), autoionization (resonance) occurs. In Eq.(45) /,m are the angular -25- momenta of the excited electron and

fa'=/' for W>WS |a'=p'for W

This shows that the Stark effect is taken into account only for the closed channels. The function H satisfies the boundary condition at large electron-core separation:

^ W>WS (48a)

H(W'a'm';Wlm)= £ G(W'a'm';W"a"m")Lra-m-tylm (49) y"a"m" or in matrix notation H=GL (50) where L is a constant matrix and G given by Eq.(41). Dividing the matrix equation (50) into open and closed channel parts,

Lo H, L, (51) boundary conditions (Eq.(48)) can be imposed. Eq.(48b) is satisfied only when the exponen- tial decreasing function in the barrier region T\b

Lo = l (52a) (]X» (52b) and

H0=V+-¥-S* (53a) 4 2i& 1 Hc=-WlC' (X«+e ]~ X'co (53b) where Wj = y sinA+y cosA (54) y and y are the Stark regular and irregular functions, respectively, given in Eq.(23). A is the WKB phase integral of the Stark bound motion given in Eq.(22) and x is given in Eq.(41). -26-

The S-matrix can explicitly be expressed in terms of A and %:

To calculate the oscillator strength (Eq.(43)) the dipole matrix element Dja is defined: (56)

D,a is divided into a resonance and non-resonance part:

3;a=3$r>+3£>= (57) x X + £

Of interest here is only the resonance part of Eq.(57). This yields [23], using Eqs.(53) and (28): 3£ [2A]"1 (58)

with

A and B are the dipole transition moments for the Coulomb regular and irregular functions independent of F and are in principle known from zero-field data. This result (Eq.(58)) substituted in Eq.(43) gives the oscillator strength per energy for the autoionizing Stark states.

4.4. Implementation To calculate the Stark effect two different types of computer programs have been used. Quasi-bound Stark spectra were obtained using the diagonalisation method presented in section 4.2. Input parameters were the quantum defects of the relevant zero-field states, the value of the field strength, the magnetic quantum number m and the ionization limit. The matrix to be diagonalised contains on its diagonal the zero-field energies (Eq.(l)) and on the off- diagonal the matrix elements of the Stark operator (Eq.(19)). The appropriate expression for barium and helium of this dipole matrix element including angular momentum coupling coefficients is given in the related chapters. The radial part of the matrix elements is calculated in the Coulomb approximation [21] following the procedure described by Zimmerman et al. [13], The size of the matrix depends on the value of the field strength and the desired -27-

accuracy. To obtain results for helium with an accuracy down to MHz level it was necessary to include five n-values in the calculation for n around 40 and F up to 35 V/cm= 1/4F, (see Chapter 5). After diagonalisation (procedure from CERN-library) the program gives the energies of the Stark levels relative to the hydrogenic zero-field level per field strength and (optionally) a measure of their oscillator strength. To calculate Stark spectra originating from autoionizing states the MQDT-formalism adapted for the Stark effect (section 4.3.5) was implemented in a computer-code. Required input parameters were the zero-field reactance matrices R° for all dipole-coupled states, the electric field value F, the magnetic quantum number m and the ionization limit. The program calculates the oscillator strength per energy (Eq.(43)) utilizing Eq.(58). To gain insight in this new method of calculating Stark spectra, only levels not influenced by perturbers (far away in energy) were studied. This reduced the multichannel problem to a two channel case. These two channels are the discrete autoionizing and its continuum channel. Values of the R° matrix elements were obtained from zero-field values of linewidths and quantum defects [24]. In general autoionizing states with / > 3 may be considered discrete (coupling with continuum is absent) and degenerate (|i=0) [25]. For this reason only input data for the R°- matrix from levels up to 1=3 are included, thus reducing the calculational work. This implies that R° is set 0 for states with / > 3. Furthermore coupling of angular momenta is neglected and the system is considered approximately hydrogenic (B is set zero in Eq.(58)). Results of calculations along these lines for the Stark effect in barium 5dnf autoionizing states are discussed in Chapter 6. It turned out that, despite the simplifications, the program still is very computer-time consuming, especially for high n. This is mainly due to the two complex matrix inversions (Eq.(42) and Eq.(58)).

Acknowledgement: I wish to thank K. Sakimoto for making available the computer-code for the MQDT treatment of the Stark effect in autoionizing states and for useful suggestions concerning the theory.

References 1. R.D. Cowan: The theory of atomic structure and spectra. Berkeley Los Angeles London, University Press 1981 2. H.A. Bethe and E.E. Salpeter: Quantum mechanics of one- and two-electron atoms. Berlin Gottingen Heidelberg, Springer 1957 3. MJ. Seaton: Rep. Prog. Phys. 46 (1983) 167 4. U. Fano and A.R.P. Rau: Atomic collisions and spectra. Academic Press 1986 5. D.A. Harmin: Phys. Rev. A 24 (1981) 2491 6. D. Park Z. Phys. 159 (1960) 155 -28-

7. J.W.B. Hughes: Proc. Phys. Soc. 91 (1967) 810 8. K. A.H. van Leeuwen: Thesis Vrije Universiteit, Amsterdam 1984 9. R J. Damburg and V.V. Kolossov: In: Rydberg states of atoms and molecules. Cambridge, Cambridge University Press 1983 10. K.T. Lu and A.R.P. Rau: Phys. Rev. A 28 (1983) 2623 11. E. Luc-Koenig and A. Bachelier: I. Phys. B: At. Mol. Phys. 13 (1980) 1743 12. E. Luc-Koenig and A. Bachelier: J. Phys. B: At. Mol. Phys. 13 (1980) 1769 13. M.L. Zimmerman, M.G. Littman, M.M. Kash and D Kleppner: Phys. Rev. A 20 (1979) 2251 14. W. van de Water, D.R. Mariani and P.M. Koch: Phys. Rev. A 30 (1984) 2399 15. D.A. Harmin: Phys. Rev. A 26 (1982) 2656 16. D.A. Harmin: Phys. Rev. Lett. 49 (1982) 128 17. D.A. Harmin: Phys. Rev. A 30 (1984) 2413 18. K. Saldmoto: J. Phys. B: At. Mol. Phys. 19 (1986) 3011 19. K. Sakimoto: J. Phys. B: At Mol. Phys. 20 (1987) 807 20. C. Fabre and S Haroche: In: Rydberg states of atoms _nd molecules. Cambridge, Cambridge University Press 1983 21. D.R. Bates and A. Damgaard: Phil. Trans. R. Soc. 242 (1949) 101 22. U. Fano: Phys. Rev. A 24 (1981) 619 23. K. Sakimoto: private communication 24. A. Giusti-Suzor and U. Fano: J. Phys. B: AL MoL Phys. 17 (1984) 215 25. E.A.J.M. Bente and W. Hogervorst to be published Chapter 3

Stark manifolds of barium Rydberg states.

ABSTRACT

In a CW laser-atomic-beam experiment Stark manifolds in barium ori- 1 1>3 3 ginating from the Rydberg states 6s40f F3, 6s40g G4, Gs and 6s40h 1>3H5 have been studied. Accurate quantum defect values for higher orbital angular momentum states (/ = 6,7) have been determined. TTie Stark manifolds are also calculated by diagonalisation of the energy matrix in the presence of an external electric field. Good agreement between experiment and calculations is obtained. -31- Atoms, Molecules and Ousters © Springer-Verlag 1987

Stark manifolds of barium Rydberg states

CT.W. Lahaye, W. Hogervorst, and W. Vassen Subfaculteit Natuur- en Sterrenkunde, Vrije Universiteit, de Boelelaan 1081, 1081 HV Amsterdam, The Netherlands

Received 1 June 1987

In a CW laser-atomic beam experiment Stark manifolds in barium originating from 3 3 U3 the Rydberg states 6s40/ 'F3, 6s40g '' G4, G5 and 6sA0h H5 have been studied. Accurate quantum defect values for higher orbital angular momentum states (/=6, 7) have been determined. The Stark manifolds are also calculated by diagonalization of the energy matrix in the presence of an external electric field. Good agreement between experiment and calculations is obtained.

PACS: 32.30.Jc; 32.6O.i

1. Introduction external electric field scales as n2, whereas its binding energy is proportional to n~2. This implies that for Over the last decade, laser-spectroscopic investiga- high n values at relatively modest field strengths Stark tions of various properties of Rydberg states in alka- manifolds may be observed. As the quantum defect, line and alkaline-earth elements, such as level ener- which measures the deviation from a hydrogenic level gies, natural lifetimes, hyperfine structure splittings structure, decreases steeply as a function of /, excita- and isotope shifts have given much insight into the tion of high /-levels will enable the observation of structure of these one- and two-electron systems [1]. hydrogenlike behaviour at even lower electric field Also the behaviour of atoms in DC electric fields has strengths. gained interest. Most attention has been given to the In this paper we report on high resolution laser quadratic Stark effect in isolated levels. In the two- experiments on Rydberg states of the barium atom electron atoms the field-induced splittings mainly in the presence of an electric field. Several high /-levels l 3 3 13 served as a sensitive probe for series perturbations such as 6snf F3, 6sng '• G4, G5 and 6snh H5 caused by configuration interaction [2, 3]. were excited, where we more or less arbitrarily se- With increasing strength of the electric field devia- lected n = 40. In all cases the field-induced excitation tions from the quadratic Stark effect and a gradual of /+1 and (+2 states in weak fields was observable, transition to a linear effect may be observed. In these enabling the determination of their quantum defects. circumstances the field energy contribution dominates To calculate the Stark structure of Rydberg states the non-hydrogenic splitting of the energy levels ac- we adapted the method developed by Zimmerman cording to the value of the orbital angular momentum et al. [4], i.e. a complete diagonalization of the energy / (for a given value of the principal quantum number matrix in a spherical basis. Other methods, based on n). The resulting Stark angular momentum manifolds Quantum Defect Theory (QDT, [5]), have been were first investigated in alkali-atoms (see e.g. Zim- worked out for alkalis, but are still under develop- merman et al. [4]). In these experiments pulsed lasers ment for earth-alkalines (Multichannel QDT) [6, 7]. and strong electric fields (up to 6 kV/cm) were used The results on the calculations of the manifolds and to excite levels with low I values (i.e. 15 p states). An- the relative intensity distribution over the levels are gular momentum manifolds upto the n-mixing region presented in Sect. 4 and compared with the observed and above the saddle point level were observed. Stark maps. Section 2 deals with the theoretical back- The coupling energy of an excited electron in an ground and Sect. 3 describes the experimental setup. -32-

2. Background J is the total angular momentum, M its z-component and y denotes all other quantum numbers. To evalu- The electric field F defines the z-direction and is as- ate this matrix element it is convenient to express sumed to influence only the Rydberg electron. The the wave function as an LS-coupled state. For two hamiltonian describing the atom-field interaction is: electron systems (barium) this reads: (1) where P is the electric dipole operator. A general matrix element may then be written as: The Schrodinger equation may be written as:

= eF<«! /, n2 l2 LSJM\r cos0\n\ /', ri2 l'2ES' J' Af'>. with W the energy and V (r) the wavefunction of the (6) system. The zero field unperturbed case (HF is zero) Under the influence of the electric dipole operator is assumed to be solved. The unperturbed level ener- only one electron is allowed to change quantum gies are given by the Rydberg formula: numbers in a transition. Here the first electron (6 s electron) is assumed to remain stationary, the field R (2) only influencing the Rydberg electron. After separat- ing the angular and radial parts and some angular / is the ionization limit, R the Rydberg constant for momentum algebra the matrix element (6) reduces the atom in question and 6, the quantum defect of to: the level with orbital angular momentum /. <5, determines the deviation from a hydrogenic level energy. (,yJM\P-F\y'J'M'y (In barium 7 = 42034.902 cm"' and R 1 = 109736.882 cm" ). (7) In approximation the radius of the electron orbit [8] is given by (in atomic units): I J I J'\(L J S)(l2 L lt\ \-M 0 M)\j' L. 1)\L 1'2 1} = 2«2 (3) From (1) and (3) it follows that the contribution of the electric field to the energy will be proportional for l'2 — l2±\ and zero otherwise and M = M', S = S'. t2mx is the largest of l2 and 1'2. The transition integral For higher /-states (I>2) with non-penetrating is defined by: electron orbits the quantum defect is small and consequently its influence on the hydrogenic level energy (8) may be considered as a perturbation. The zero field hydrogenic level energy in first order approximation where Rnl{r) is the radial wave function of the nl- may be expressed as electron. W j = {n-z + 6,n-3. (4) For Rydberg states the radial integrals may be n calculated in Coulomb approximation, assuming gen- It follows that the contribution of the field to the eralized hydrogenic radial functions which are nor- total energy is much larger than the quantum defect malized to the experimental level energies [9]. Best term when § F n2 P 5, n ~3. Under this condition a level results give a simultaneous integration of this Bates- with orbital angular momentum / may be assumed Damgaard approximation and equation (8) [10]. to be degenerate with all higher /-levels and linear To reproduce the overall intensity profile of a Stark effect", are observable. Stark manifold it is also necessary to calculate the The on,) non-zero elements in the energy matrix transition probability to excite a Stark state from a fall on the diagonal and two off-diagonals. The diago- low lying state. The oscillator strength for a transition nal elements are the zero field level energies, calculat- from a ground state W, J, M to a Stark state W, ed with (2) from known quantum defects. The off- AT is [II]: diagonals contain the electric field contribution and these matrix elements have the form:

l (9) (5) -33-

bration experiment on the polarizabilities of the 6 s 18 d 30 and 6s 20d 3D levels of calcium [9]. Excita- tion spectra are recorded by monitoring the signal of the electron multiplier as a function of laser frequency. To calibrate the spectra part of the visible light output of the dye laser is sent through a confocal Fabry-Perot interferometer (free spectral range appr. 150 MHz) which is length-stabilized by locking it to an iodine stabilized HcNe laser. Both the signal from the electron multiplier and the calibration interferometer are stored on a microcomputer which also controls the laserscan. Laserscans of up to 30 GHz, involving 2048 computersteps, can be made. The HeNe laser also serves as a reference for wavelength measurements, which are performed with a fringe- Fig. 1. Schematic of the experimental setup. EM, electron multiplier; counting Michelson interferometer (accuracy better PM, pholomultiplier; WP, wollaston prism; PD, photodiode than l(T4nm).

3 b. Lasers where V^-j- is the unitary transformation which diagonalizes the Stark matrix. The 6snf 'F3 levels (« = 28^t0) are populated from U3 Although there is no selection rule for n in an the 6s 5d O2 metastable states using an intracavity electric field it was sufficient to include only one n frequency doubled, actively stabilized CW ring dye value in the calculation of the Stark maps for the laser (Spectra Physics 380 D, dye Rhodamine 640). moderate field strengths used in the present experi- Both temperature-tuned ADA (Ammonium Dihydro- ment. A program version adapted to include three gen Arsenate) and angle-tuned LiIO3 (Lithium lo- neighbouring n values is available and comparisons date) non-linear crystals have been used to produce have been made. UV radiation. The 6s40g states are reached via two step excitation [12]. For the first step a linear dye laser (Spectra 3. Experimental setup Physics 580) is used and locked to one of the atomic transitions 6s5d->5d6pina separate interaction re- 3a. Atomic beam production and detection gion (see Fig. 1). The ringlaser then scans the Il3 3 5d6p-»6s4Og excitation. To excite the G4, G5 The experimental setup is shown schematically in l states the 5d6p F} (A = 648 nm from 6s 5d 'D2, dye Fig. 1. A collimated beam of barium atoms is perpen- 3 Kiton red) respectively F4(,l = 706nm from 6s 5d dicularly intersected by the light of one or two dye 3 D3, dye Pyridine 1) as intermediate levels have been lasers. The beam is produced by heating a tantalum used, whereas the ring laser operated on the dye oven containing a natural sample of barium with a DCM respectively Rhodamine 110. With a ringlaser tungsten filament positioned a few mm in front of power of 100-200 mW no saturation effects in the and above the orifice. Metastable 6s 5d states at ap- 1 5d6p-*6s4Qg transitions were observed. No effort proximately 10000 cm" are populated by running was made to induce manifolds via the iG level as a discharge between filament and oven through the i 2 this level may only be weakly excited [13]. atomic beam. The metastable 5d 'C4 level at 13 1 The 6 s 40 A HS states are excited in one step 24696.278(6) cm" [12] is populated directly by the 2 hot filament. from the metastable Sd 'G4 level [12]. The ringlaser The atomic beam is collimated with an adjustable operated on the dye Rhodamine 6G and a power slit. The population of the Rydberg levels is detected of 200-300 mW did not saturate this transition. a few mm downstream from the excitation region by All lasers are frequency stabilized on external eta- field ionizing the Rydberg atoms and counting the Ions, yielding band widths in the order of 1 MHz. detached electrons with an electron multiplier. The excitation region is centred between two small 3 c. Energy values and intensities capacitor plates, which sustain the applied electric field. The separation between the plates is measured Accurate values of the different zero field level ener- to be 6.03(6) mm. This value was confirmed in a cali- gies have already been measured [11, 12]. Positions -34-

of the various manifold peaks are measured relative to these zero field levels utilizing the Fabry-Perot calibration spectra. The intensity of a peak is the integrated signal concentrated in the centre of a resonance.

4. Results and discussion

4 a. Determination of quantum defects of higher l-levels In the transition from quadratic to linear Stark effect the wave function in the presence of a weak field (n = 40; F<0.5V/cm), \yJM)F may be expanded as t a linear combination of the zero field wave functions of the contributing levels:

(10) FREQUENCY •/•.'• with the expansion coefficients given by -<.yJM\P\y'J'M> (11) yj-*-iJ W° is again the zero field level energy (see (2)) and a is the normalization factor for the excited zero field wave function. Formula (10) represents in first order the field dependent admixture of opposite parity levels into the wave function of the sublevel \yJM~). The contributing opposite parity levels \y'J'} will in turn have an admixture of the level \yjy in their wave functions and, depending on the magnitude of the expansion coefficients (11), may therefore be excited in the presence of a weak electric field. In a first experiment the quantum defects of the, ll3 at that time not directly observed, (ssng G4 states of barium were determined. Commonly in weak fields no linear Stark effect is observed in excitations to *— FREQUENCY the 6sn/Rydberg levels, but due to a perturbing in- l Fig. 2 a and b. Spectra of 6s 40/ 'F in a weak Field, a Excited teraction with the 5dSp F} level [14] the quantum 3 1 from the metastable 6s Sd 'D2. b Excited from the metastable 6s 5d defects of the 6snf F3 levels for n>28 are nearly 3 l 3 D2- Not assigned peaks stem from hyperfine structure zero. Although in later experiments the 6sng - G4 series were directly excited [12] the electric field induced population of a G-level gives a decisive answer whether the level has dominantly triplet or singlet l is also observed as 'G4 contains a small triplet admix- character. In a first experiment 6snf F3 (n = 28-4O) l ture [13]. This identification is of importance for the were excited using 6 s 5d D2 as ground state. As 6s 5 d MQDT analysis of the 6sng series [13], as it shows metastables are nearly pure LS-coupled states [11] the interesting feature that the singlets have slightly only singlet character is excited. The experimental lower energy than the triplets. The quantum defects 3 l 3 spectrum only shows the 6sng 'G4 level (n=40, of the '• G4( G45 = 0.056, G4 5 = 0.052) for n = 28- Fig. 2 a). Next an experiment using the metastable 3 40 deduced in this way have been used by Zaki et al. 6.i 5 d D2 level was performed. Then the small frac- l [15] to confirm their predictive calculations of the tion of triplet character in F3 [14] is excited and 3 6sng J = 4 series from a study of the quadratic Stark the experimental map (Fig. 2 b) displays G4 (relative- effect in 6snf series. These results are also in good ly strong peak) and 'G4 (weak peak). This latter peak agreement with values found by Gallagher et al. [16] -35-

for lower n-values taking into account the observed energy dependence [13]. a) 1 n = 40 10 Due to the weak coupling via the G states (rela- F=0.15V/cm tively far away) signals from levels with l>4 could not be observed in this weak field case. In fact above IGHz a certain threshold field strength the complete mani- 3s« fold appears. Attendent difficulty in the observation of these manifolds is the hyperfine structure of the odd barium isotopes which obscures the onset of the manifold. For this reason this method of exciting manifolds was not pursued further. Next in a two 'H= 3 3 step excitation process the 6s 40g ' • G4 and G5 levels have been excited. In the first step only the even isotope 13SBa (abundancy 71.7%) is excited from the t metastable 6s 5d levels. In the second step the 6s40g levels of only l38Ba are populated, so hyperfine structure peaks are not present. As a consequence of smaller energy separations, • FREQUENCY and thus stronger coupling with the field (see (11)), higher / values (i>4) are easily observed in the weak b) field region (f<0.5 V/cm). Also the field-induced ex- n=40 citation of 'f3 is visible in the spectra. Extrapolation F=0.15Wcm of the various energy separations as a function of 3 the field strength relative to the unperturbed zero field 1GHz peaks (6s ng level energies given by Vassen et al. [12]) 3 results in quantum defects off/-, /- and via G5, even K-states (see Fig. 3 and Table 1). Table 1 suggests that 3 '//., and H6 coincide (« = 40). Vassen et al. [12] in If3 a direct excitation studied the //5 Rydberg series, including hyperfine structure and their analysis of the hyperfine data [13] provide information on the posi- 3 i tion of the H4 and Hb levels. A preliminary analysis t l 3 i 3 confirms that //s and tf6 (as well as H4 and H5) X coincide. The deduced quantum defects of the high H /-levels (/ = 6 and 7) follow roughly a T5 law [17]. uu Hm VJI I mi ,... Gallagher et al. [18] measured ng—nh — ni — nk fre- r quency intervals in a combined pulsed laser- and rf- FREQUENCY experiment for n values around 20. Extrapolating these values to n = 40 gives reasonable agreement with C) the present results. n = 40 I-3 F=0.15V/cm In a consecutive experiment H5 (and 'F3) have z been populated directly from the metastable 5d 'G4 3 level. Again hyperfine structure of the '- H5 and 'F3 1 | 1GHZ | levels is blurring the onset of manifolds. Our aim was to observe even isolated peaks of 1>~I. Figure 3c shows this was not possible. Only a single K-peak z 'F 3 or the onset of the complete manifold was observed aJ in weak fields. Obviously for states with 1>1 the de- Z generacy hampers resolved observation. •> t

Fig. 3a-*. Examples of the excitation of higher /-levels in the pres- 1J J ence of an electric field. » Via 6s Wg C4. b Via 6s40g GS. l 5 cVia6s40A - tf< FREQUENCY -36-

Table 1. Quantum defects of high J-states deduced from low-Held excitation of ''3G ,3C and IJH levels. *direcl excitation n.40 4 5 5 F . 3B6 V/cm IMI .3 A via G viatf

'H, 0.0187 (2) 0.0187 (2)* 0.0195 (2) 0.0197 (2)* 'H, 0.0187 (3) _ '\ 0.0092 (5) 0.0090 (5) 'I, 0.0092 (5) - K,. = 0.004 =0.004 B 0

4 b. Energy positions of manifold levels 4— FREQUENCYit Fig. 4. Stark manifold originating from 6s 40g ' G Although manifolds were studied in excitations of the 4 3 3 '• G4-, G5- as well as '^Hs-levels for reasons of similarity only one type of manifold will be discussed, l l namely the one observed via the 6s40g 'G4. The with 'G4 (via D2 and F3) is weak. As 'P, crosses levels of the "H- and G5-manifold" split into 6 sublev- the manifold at every junction an anticrossing occurs. els (according to the quantum number |iW|) against For the first anticrossings (at field strengths around 5 in the manifold excited via the G4-states, making 3 V/cm) calculations show a shift of these levels of the last one less complicated. The manifolds via triplet about 50 MHz. This could not be verified experimen- states are not discussed as in the calculations the fine tally because both 'Pt and nearby manifold levels structure has to be taken into account, only hamper- are not observable at such low values for the field ing the analysis. strength. At higher field strengths the anticrossing gap 3 In the two step excitation of 'G4 also G4 is visible increases and 'P, acquires increasingly more mani- (see e.g. Fig. 3a). However its intensity is weak com- fold character (hydrogenic) until about F=7 V/cm l pared to the 'G4-signal, the triplet manifold appear- where the level no longer can be assigned P,. For ing only as a weak background in the singlet mani- F=8.02 V/cm an experimentally observed shift due fold. In Fig. 4 an experimental map of a manifold to this anticrossing occurs (at around — 3 GHz). for F = 3.86 V/cm is shown. For this field strength However not all the levels could be detected in this the |JVf| = 4 component of 'G4 has already merged region because of low intensity of manifold peaks with the manifold in contrast with the other sublevels around the anticrossings (see sect. 4c), level broaden- which are still observed separately. For reasons of ing (up to 100 MHz) and overlap of different \M\ sub- clarity the analysis and discussion is limited to the levels hampering assignment. behaviour of the M=0 and |M|=4 sublevels. The The |M|=4 manifold in Fig. 5 b shows a more other sublevels will show an intermediate behaviour hydrogenic pattern as lower /-levels (/<4) with pro- compared to these outermost sublevels (see e.g. nounced quantum defects do not contribute in this Fig. 6). case. To calculate the manifold level energies only In weak fields the behaviour of "isolated" lev- quantum defects of all I levels (40 values) are needed. els will not be linear but quadratic. Figure 6 shows The first eight values (/ = 0, 7) are determined experi- the energy shifts of the |JW|-sublevels of 'G4 as a mentally [12, 14, 19, 20 and this paper], the others function of field strength, exhibiting excellent agree- via a I'' law [17]. In Fig. 5a and 5b calculated mani- ment of experiment and calculations. Using equa- folds using the procedure outlined in sect. 3 b as a tion (4) for 40 'G4 a linear Stark effect should be- function of the field strength are shown for resp. M = 0 come apparent at a field strength F % 1 V/cm. Clearly and |M| = 4. For reasons of clarity the experimental resolved manifold peaks were observed at a field points of the manifold levels are inserted only for strength F*0.9 V/cm confirming a smooth transition F = 8.02 V/cm, but all other data points also coincide to the linear Stark effect. with the theoretical curves. The 6s43rf 'D2 level start- In the hydrogenic case the mean spacing between ing at +29.8 GHz in Fig. 5 a falls outside the range neighbouring manifold levels in first order perturba- of the graph. The dotted lines represent 65 44s 'So tion theory is 3Fn (independent of the other quantum and 6s44p 'P, (starting at -22.8 GHz and numbers [8]). Determination of the actual spacing —8.9 GHz respectively), not observable in the experi- AW as a function of the field strength gives a more mental spectra because the coupling of 'Pi and 'So profound and surveyable way of comparing theoreti- -37-

Fig. 5a and b. Calculated sublevel energies as a function of the electric Held strength, •experimental ? 4 6 points, a M = 0 manifold, b |A/|^4 manifold —• FlWcm I

Fig. 6. Calculated splitting and shift of 6s 40g *G4 in the presence of an electric field, • experimental points cal and experimental results instead of inspecting the bation theory. The level spacings increase near the position of each individual spectral component. In edges of the manifold, especially for the Af=0 compo- Table 2 averaged values of AW as a function of F nent. The second order contribution only yields a dif- are given for M=0 and |M| = 4. It also includes the ference of + or -0.5 MHz at F- 8.02 V/cm for resp. hydrogenic behaviour according to 3Fn. Notice that the upper and lower level spacings compared to the the spacings for barium are slightly larger than for level spacings in the middle of the manifold not ex- hydrogen. The errors are standard deviations stem- plaining these differences. Also the positions of the ming from the fact that the spacing between manifold levels and hence their spacings do not change in an levels is not constant as predicted in first order pertur- observable way for the experimentally used moderate -38-

Table 2. Mean energy separations IMHz) between manifold sublevels for different field strengths. * Calculated with the extended program version using three neighbouring n-values [n = 39, 40, 41)

/'|V cm I 3 Fit

0.9} 165 125) 159 (7) 165 (25) 152 (1) 143 1.10 185 (29) 182 (1) 185 (29) 177(1) 169 1.27 206 (15) 205 (4) 206 (15) 203 (1) 195 1.44 234 (12) 231 (51 234 (12) 230 (1) 221 1.60 259 (19) 257 (6) 259 (19) 255 (1) 245 1.76 287 1141 285 (8) 287 (14) 2S1 (1) 270 2.10 340 (19| 335 (8) 340 (19) 331 (6) 322 2.60 425 (27) 417 (10) 427 (22) 411 12) 398 3.10 510 (22) 501 (15) 501 (18) 487 15) 475 3.93 645 (29) 634 (21) 637 (23) 613 (13) 602 K.02 1273 126) 1284 (38) 1243 (19) 1239 (21) 1229 1282 (421* 1239 (28)*

field strengths when adjacent manifolds (n = 41, 39) are included in the calculation. Hence the variation in spacing stems from the fact that the system is not completely hydrogenic for the applied field strengths. The levels with low / (/<5) are still not completely merged with the manifold. This effect is more pronounced for the M = 0 than for the | M\ = 4 component (see Table 2) because the latter does not contain levels with l<4 with a pronounced quantum defect. Al- though the spacings over a manifold are slightly varying the experimental values are well reproduced by the calculations. Another feature of the recorded manifolds is the fact that outermost manifold levels ar broadened 16 12 8 4 0-4 -8 -12 -16 compared to the centre ones preventing resolved observation of different \M\ sublevels. Assuming the manifold to be purely hydrogenic in first order pertur- b) /I. bation theory the level energies are given by W= n.40 2 F= 3.86 V/cm / — l/n + l Fnk, here k=nl —n2 with «, and n2 para- 1 Ml =4 / bolic quantum numbers [8], It follows that small stray fields AF broaden the upper and lower levels more than the middle ones. Comparing the linewidths of the field broadened upper levels to the widths of the narrow middle levels for F = 8.02 V/cm yields a value for the stray field of only AF st 10 mV/cm. This also happens to be the experimental error. The broadening of the outermost manifold levels thus may be ascribed to the presence of such small field inhomo- geneties. I 1 ....UT-*"T 1 1 1 1 1 16 S 0 -4 -8 -12 -16 4c. Relative intensity distribution over a manifold 4— W(GHz) Fig. 7« and b. Relative intensities of the manifold excited via 6s 40g ] A remarkable feature of Fig. 4 is the intensity profile, C4 for F=3.86 V/cm. x experimental, • calculated; the curve only especially in the right wing of the manifold. A closer serves (o guide the eye. a JW=0. b | M| = 4 inspection reveals that the high intensity peaks here stem from the |A/|=4 component of 'G4. In Fig. 7 cal values is to guide the eye and gives insight into the M = 0 and |A/|=4 components derived from the the intensity envelope of the manifold. These profiles experimental map at F=3.86 V/cm (Fig. 4) and their are quite different for the two \M\ components: the (integrated) intensities compared with values calculat- |A/|=4 component of the 'G4 already merges with ed with (9) are given. The line connecting the theoreti- the manifold, resulting in the steep rise in intensity -39- towards lower energies (Fig. 7 b), whereas the M=0 the linear Stark effect offers a good method to estab- component is still observed resolved (see Fig. 4). This lish their energy positions. brings about the sharp intensity maximum near The onset and development of manifolds and the — 13.8 GHz in addition to a smooth intensity distri- splitting of each level into its sublevels according to bution over the manifold with a minimum in between \M\ have been studied. The results are compared with l at approx. -4.3 GHz (Fig. 7a). The 'So and D2 lev- theoretical calculations based upon a diagonalization els fall outside the range of the figure and have nearly of the complete energy matrix. There is excellent zero intensity. Also 'P, is visible in the calculated agreement between experiment and calculations upto curve at —7.15 GHz having an intensity of approx. moderate field strengths (F< \FC; Fc saddle point field 1 % of the M = 0 ' G4 peak. This low value in combina- strength). Also the relative intensity distribution over tion with the presence of other \M\ sublevels explains the manifold sublevels is studied and can be repro- why the 'P, is difficult to observe experimentally. The duced quite accurately in the calculations. intensity envelopes show good agreement of experi- It may be concluded that upto moderate field ment and calculations. However the experimental strengths the applied method of diagonalization of points tend to scatter around the calculated curves, the energy matrix still holds to explain the observed especially for narrow peaks in the middle of the mani- manifolds. For higher field strengths as well as for fold where the linewidth becomes comparable to the autoionizing states where interaction with the contin- stepsize of the laserscan (about 15 MHz). uum has to be included alternative approaches (based Some experimental peaks are missing in the M = 0 upon MQDT) should be followed. Experimental stu- graph (Fig. 7a) because of almost zero intensity (near dies of the Stark effect in autoionizing series in barium — 4.3 GHz) or lack of resolution due to level broaden- are in progress in our laboratory. ing (near —9 GHz). The experimental values on the steep rising part of the |M|=4 curve (Fig. 7b) are The authors are indebted to Erwin Bente, Stephen Lorie and systematically larger than the calculated values. This Jacques Bouma for their assistance in running the experiment and Hans Wijnen for support with the computer program. Financial overestimation is due to contributions of other \M\ support from the Foundation for Fundamental Research on Matter sublevels not spectrally resolved from the |W| = 4 (FOM) and the Netherlands Organisation for Advancement of Pure component in this region, e.g. compare points at Research (ZWO) is gratefully acknowledged. -10.4 GHz in Fig. 7a and 7b. Higher field strengths up to 150V/cm (saddle point field strength for «=40: F «125 V/cm) than re- References ported here have also been applied. A dense spectrum 1. Stebbings, R.F., Dunning, F.B. (eds.). Rydberg slates of atoms of broad resonances and narrow peaks is observed. and molecules. Cambridge: Cambridge University Press 1983 These spectra cannot be explained within the frame- 2. Van Leeuwen, K.A.H.: Thesis, Amsterdam 1984 work of the present method of calculation as diagona- 3. Zaki Ewiss, M.A.: Thesis, Amsterdam 1985 lization of the energy matrix only predicts energy 4. Zimmermann, MX., Littman, M.G., Kash, M.M., Kleppner. D.: values and no Iinewidths. Moreover the calculation Phys. Rev. A 20, 2251 (1979) 5. Harmin, D.A.: Phys. Rev. A26, 2656(1982) becomes too cumbersome and the matrix too large 6. Harmin, D.A.: Comm. At. Mol. Phys. J5, 281 (1985) considering the many levels which have to be in- 7. Sakimoto, K.: J. Phys. BI9, 3011 (1986) cluded. Also the continuum, having important influ- 8. Bethe, H.A., Salpelcr, E.E.: Quantum mechanics of one- and ence at high field strengths, is not accounted for in two-electron atoms. Berlin Gottingen Heidelberg: Springer 1957 this method. Incorporation of the Stark effect in the 9. Bates, D.R., Damgaard, A.: Phil. Trans. R. Soc. 242, 101 (1949) (0. Van Leeuwen, K.A.H., Hogervorst, W.: J. Phys. B16, 3873 (1983) framework of MQDT [5, 6, 7] seems to be the only 11. Cowan, R.D.: The (heory of atomic structure and spectra. Berke- solution to overcome these problems. ley, Los Angeles, London: University of California Press 1981 12. Vassen, W., Bente, E.A.J.M., Hogervorst, W.: J. Phys. B 20, 2383 (1987) 13. Vassen, W., Hogervorsl, W. (to be published) 5. Conclusions 14. Post, B.H., Vassen, W., Hogervorst, W., Aymar, M., Robaux, O.:J. Phys. B18, 187(1985) Stark manifolds in barium have been studied in exci- 15. Zaki Ewiss, M.A., Hogervorst, W., Vassen, W., Post, B.H.: Z. tations o(6snl Rydbergstates (n=40) using CW laser Phys. A - Atoms and Nuclei 322, 371 (1985) spectroscopy. Exciting high orbital angular momen- 16. Gallagher, T.F., Gounand, F.. Kachru, R., Tran, N.H., Pillet, 3 3 P.: Phys. Rev. A27, 2485 (1983) tum states (i.e. 'F3, '- G4, G5 and '^//s) the linear 17. Fabre C, Haroche, S.: In: Rydberg states of atoms and mole- Stark effect becomes apparent even at weak fields cules. Cambridge: Cambridge University Press 1983 (Fts 1 V/cm) because of small quantum defects of the 18. Gallagher, T.F., Kachru, R., Tran, N.H.: Phys. Rev. A26, 2611 zero field states (<5| < 0.06). In first instance quantum (1982) 19. Aymar, M., Camus. P., Dieulin, M., Morillon, C: Phys. Rev. defects of higher ( states (1=6, 7, 8) have been deter- A18, 2485(1983) mined. For these not directly accessible high / states 20. Aymar, M., Robaux, O.: J. Phys. B12, 531 (1979) Chapter 4

Electric field induced avoided level crossings in bound Rydberg states of barium

ABSTRACT

In a CW laser experiment on bound Rydberg states of barium electric- field-induced level crossings were studied. A regular type of avoided crossing 3 3 of 6snf F2 and 6s(n+4) Sj levels (n=60, M=0) was observed under circumstances where the minimum separation was larger than the linewidth involved. This type of crossing could be reproduced using a diagonalisation procedure of the total energy matrix in the presence of the field. However, the in case of avoided crossings of the 6snf3F2 and (Ssfa-Mip'Pi sublevels, where the minimum separation was smaller than the linewidth of the "F2 level, an interference-like profile was recorded. An important feature in the explanation of this interference-like profile turned out to be the broadening of the 3F2 level by small field inhomogeneities. -43-

1. Introduction In the course of a comprehensive study of bound Rydberg states of barium in our laboratory Eliel [1] observed puzzling line profiles in the simultaneous excitation of 6s65f 3F2 and 6s69p xPj levels from the metastable 6s5d 3Dx state in the presence of an electric field. In this paper we present results of a more detailed investigation and we will show that these phenomena are induced by small electric field inhomogeneities. Electric field-induced pseudo-crossings between angular manifold components in atomic systems have been observed frequently in experiments with pulsed lasers [e.g. 2,3,4]. This type of level crossing spectroscopy nowadays is a classical method for obtaining detailed information about highly excited states [4]. For example, data on fine and hyperfine structure, Stark and Zeeman shifts, oscillator strengths and lifetimes have been collected using this method [5]. Furthermore, avoided crossings play an important role in dynamical processes such as the behaviour of an atom in (strong) time varying fields [3,6,7], An avoided crossing is a consequence of a (repulsive) interaction coupling the levels involved. The resulting mixed wavefunctions strongly influence e.g. atomic ionization rates [8,9] when compared with systems in the absence of the interaction (e.g. the hydrogen atom in an electric field [10]).

In this paper we report the result of a study in a high resolution experiment, described in section 2. The observed interference-like behaviour between the 6snf 3F2 and 6s(n+4)p *Pi levels in the presence of an electric field is discussed in section 3.3. The more common known case of avoided crossing [3] is also observed when the 3Si state, excited in the electric 3 field, approaches the F2 level. This phenomena is analysed in section 3.2.

2. Experimental setup and results A collimated beam of discharge-populated metastable barium atoms was perpendicularly intersected with the light of an intracavity frequency-doubled Rhodamine 6G ring dye laser (bandwidth <, 1 MHz). The ultraviolet radiation was produced using a temperature-tuned ADA non-linear crystal. The excitation region was centred between two small capacitor plates sus- taining an applied electric field. The separation between the plates was measured to be 6.03 (6) mm. The excitation of Rydberg atoms was monitored by counting the electrons, detached downstream by field ionization, with an electron multiplier. This signal together with the calibration signal from a 150 MHz Fabry-Perot etalon was stored on a microcomputer which also controlled the laserscan. A more detailed description of the experimental setup is given elsewhere [1,11]. 3 1 The 6snf F2 and 6sn'p Pi (n'=n+4, triplet fraction) levels around n=60 were directly excited from the metastable 6s5d 3Dj state. As a consequence of a perturbation the high-n 6snp *Pi levels [12] are shifted towards small quantum defect values, facilitating the excita- 3 tion of both F2 (quantum defect 8=0.155, n=60) and 'Pj (5=4.215, n=64) in one limited -44-

3 l laserscan of about 2 GHz (separation F2 <-> P\ minimal for n=60). The calculated value of 3 1 the polarizability of the F2 level is about 100 times larger than the polarizability of the Pi level. Both states shift towards lower energies and as a consequence the level separation decreases with increasing field strength F (see Fig.l). In the presence of an electric field the 3F2 level broadens and splits into its IMI components. There was hardly any effect of the 1 applied field strengths of up to 1 V/cm on the Pi le/el.

F=0Wcm F=0.66 V/cm jo F= 0.795 V/cm 1 GHZ 350 MHz

1/5

IM 1= 1

136

F=0.91 Wcm F= 0.98 V/cm F- 1.10 V/cm 350 MHZ 750 M Hi 750 MHz

F = 1.18 Wcm F = 1.30 V/cm 750 MHz 750 MHZ

3 3 Figure 1: Spectra recorded for the one step laser excitation 6s5d Di -» 6s60f V2, 6s64p 'Pj for various field strengths F. In the upper left spectrum (F=0 Vlcm) for3^ signals of both even isotopes C138Ba and 136Ba; 3Si separation the3Sj peak gained intensity until, for the M=0 components, the two peaks had comparable linestrength and were at a minimum separation. With a further increase of field strength the intensity of the peak at the high frequency side dimin- 3 3 ished and the separation between F2 and Si grew again (compare Fig.l: F=1.18 V/cm and F=1.23 V/cm). This is a typical example of an avoided crossing. Such a crossing is also expected for the !MI=1 components but could not be observed. It was obscured by the 3 3 IMI=0,2 components of F2. Also the IMI=1 component of Sj compared with the M=0 com- 3 ponent rapidly broadens approaching the F2 level. In the last picture of Figure 1 (F=1.30 V/cm) the *Pi level as it is crossing the manifold 3 3 is shown. Also included is the still isolated F2 state. The Si level is no longer observable.

3. Discussion

3.1. Theory The origin of avoided crossings is a perturbation which breaks the symmetry of the unperturbed system thus removing the degeneracy at the crossing. Consider two states approaching each other with increasing field strength F. The electric field will mix opposite parity states into the original wave functions. Whenever there are common admixtures in then- wave functions second order perturbation theory predicts a repulsion between the states. So the minimum separation Wo is a measure of the coupling between the states. In Figure 2 such an anti-crossing is schematically shown. The eigenfunctions of states in an avoided crossing (IA> and IB>) can be expressed as a linear combination of the unperturbed (far from the crossing) eigenstates (ll> and I2>). The relevant intensities may then be written as [3]:

1,4 = Vi [(1 - cos8)Ii + (1 + cos0)I2] + sin0 -^Ixl2cos^ (la)

h='/J [(1 +cos0)Ii +(1 -cos8)I2] -sine ^I^cos*)) (lb -46-

with

tan6=- 0<6

Here V is the interaction lifting the degeneracy at the crossing, I the intensity and W the energy (see Fig.2).

Figure 2: Energy as a function of field strength for a two level system. Wo if the minimum separation at the

crossing field strength Fc. Solid lines: Avoided crossing, Dashed lines: Unperturbed energies

3.2. Avoided crossing 3F2 <-> 3Si 3 3 3 In the case of the F2 <-> Si avoided crossing only the odd-parity F2 level may be directly excited from the 6s5d3Dj metastable state. The even-parity 3Si level may be excited in the field through admixture of odd-parity character in its wavefunction. Li Figure 3 3 3 the experimental energy separation between the M=0 sublevels of F2 and Si as a function of field strength is given. Also the intensity of the low frequency peak is plotted relative to the high frequency peak, which is normalized to unit intensity (straight vertical lines). The experimental intensity of a peak is the integrated signal concentrated in the centre of a resonance. 3 3 As dominantly F2 character in the wave functions of both states is excited the growth of F2 character in the wavefunction of the 3Sj state with increasing field strength up to the middle of the avoided crossing (Fc = 1.202 V/cm) may be observed. Here the intensity is equally dis- 3 3 tributed in agreement with Eq.(l) where ll>= I F2>, I2>= I SI>. A further increase in field strength restores the situation as if the two levels actually have crossed. In Table 1 the experimental value of the minimum separation at the crossing field strength Fc is given. -47-

400-

Figure 3: 3 3 The avoided crossing 6s6O}" F2 <-> S1 Straight vertical lines: Intensities of lower frequency peak relative to the high frequency peak, which is normalised to unity strength. Solid lines: Experimental -400 Dashed lines: Calculated

To reproduce the experimental results also a calculation was performed using a procedure where the complete Stark energy matrix is diagonalised [2,11]. The coupling for M=0 3 3 3 3 of Si with F2 in the field occurs via the intermediate couplings Po <-> Dj or 3P2 <->3Di,3D3. Calculation of the actual dipole moments showed that contributions 3 3 involving P2 are much smaller than the one with P0 (about a factor of 10). For this reason contributions with3Po as intermediate level were accounted for in the calculations. Further- more only the first 8 (up to 1=1) angular momentum levels were incorporated in the calcula- 3 tion. In the actual situation not only F2 but also P states are excited. Therefore an excitation 3 3 probability for the Po level of about 1% of F2 (see Post et al. [12]) was necessary to reproduce the experimentally observed relative intensities around Fc. The zero-field energies 3 3 3 3 3 of Si, P0, Dj, G3 and IU were calculated with the Rydberg formula applying quantum 3 defects from MQDT analyses [12,13,14,15]. The energy of F2 was calculated directly from 3 3 measurement of the wavelength in this experiment The zero-field energies of I5 and Kg were calculated using the quantum defects of I and K levels given by Lahaye et al. [11]. -48-

Table 1: Experimental and calculated values of the minimum separation Wo (in MHz) of the avoided 3 3 crossings^ <-> S] and F2 <-> 'Pj

3 3 3 F2<-> S, | F2<->'Pi M=0 IMI=1 Experimental 256(5) 18(3) 33(3) Calculated 259(5) - -

Fc (V/cm) 1.2Q2 .800 .900

Results of the calculations are plotted in Figure 3 (dotted lines) and Wo is given in Table 1. 3 The error in Wo was deduced from a variation of the quantum defect of F2 (5=0.155) by 0.002 which corresponds to the uncertainty in the wavelength measurement (0.002 cm"1). Table 1 shows excellent agreement between experiment and calculations for the

minimum separation Wo. Still there is a difference between the slope of the experimental and the calculated curves (Fig.3). This deviation may be explained by noticing that only a limited number of level couplings is incorporated in the calculation. As the observed avoided crossing pattern strongly resembles the behaviour of a two level system (compare Figs.2 and 3) this probably works well for the minimum separation. However, for the slope of the two approaching levels all levels involved should be accounted for in the diagonalisation. This is not feasible as for the calculation of all possible dipole couplings level energies are not known sufficiently accurate. A deviation between calculated and experimental relative intensities was observed for field strengths larger than 1.2 V/cm. Here the high frequency peak coincides with the IMI=1 sublevel of 3F2 (see Fig.l F=1.23 V/cm), resulting in an overestimated intensity content of this peak and a smaller ratio for the relative intensities. The avoided crossing is only observed for the M=0 component. Calculations show that the dipole matrix elements of the relevant 1MI=1 sublevels are about a factor of 3 smaller than the M=0 elements. This prohibits the experimental observation of the IMI=1 avoided crossing, 3 which, in addition is obscured by the excitation of IMI=0,2 components of F2.

3 3.3. Avoided crossing F2 <-> *Pi 3 3 In the case of the F2 <-> *Pi avoided crossings the F2 level was directly excited fir - the metastable3Di state as was the'P! level via an admixture of 3Pj-character in its wavefunction [12]. Scanning through an avoided crossing one expects that signals show intensity changes as observed for the case discussed in section 3.2. In the avoided 3 l 3 crossing F2 <-> ?i this was not observed experimentally as the F2 level is broadened due to field inhomogeneities. Actually a summation of intensities over a range of field strengths determined by this field inhomogeneity was observed. Performing such a summation for a -49- hypothetical case interference-like profiles such as shown in Figure 1 (F=0.795 V/cm, F=0.91 3 V/cm) could be constructed. However, this was not feasible for the actual F2 <-> *Pi crossings as energy levels and singlet-triplet mixing coefficients, required for accurate calculations, are not sufficiently well known from present MQDT analyses for n around 60. In an earlier experiment small field inhomogeneities of about 12 mV/cm were found for the capacitor used in the present experiments [11]. In combination with the calculated high 3 polarizability this leads to a field dependent F2 level-linewidth proportional to F (HWHM 3 T=73.3*F). Experimentally 73.6*F was found showing that the broadening of F2 is completely due to the inhomogeneity of the field. Coupling with the continuum is still absent at the low values of the field used in the experiment. Obviously the broadening of the 3F2 lines is larger than the minimum gap Wo for this avoided crossing, in contrast with the 3F2 <-> 3Si case discussed in section 3.2. As an exact calculation as in the case of the 3 3 avoided crossing F2 <-> Si was not feasible a fitting procedure was developed to reconstruct the avoided crossing 3F2 <-> *Pi from the experimental data. The total spectrum was constructed by calculating

Stot(.W)= j SF(W)GFdF (2)

Fm-'AAF

Here Fm is the centre field strength, AF the field uncertainty, Gp a weighting factor and Sp a single spectrum at a certain field value. The form of Gp was assumed to be Gaussian around the field strength Fm. The actual calculation was performed using a numerical evaluation of the integral in Eq.(2). A single spectrum Sp was constructed by assigning the energy positions and intensities of levels as given by Eq.(l) with linewidths taken from zero-field data.

Furthermore, the relative intensities i! and I2 of Eq.(l) were obtained from zero-field data. A starting value for 9 was generated from an estimation of Wo from the width of the dip in the recorded spectrum (see Fig.4A and Fig.5) and from the slopes at which the levels approach using the calculated polarizabilities [16]. Also the IMI components not involved in the avoided crossing were included to obtain a good fit for the wings of the lines of the IMI component of interest. Results of fits are shown in Figures 4 and 5 and values obtained for Wo are given in Table 1. Extremely narrow crossing gaps Wo of 18(3) MHz and 33(3) MHz for the M=0 respec- 3 tively IMI=1 case were deduced, implying there is almost no coupling between the F2 and *Pi levels in the field, or, differently put, the common fraction in their wavefunction is 3 nearly zero. From the zero-field separation between F2 and *Pi this common fraction producing the repulsive behaviour at the crossing is estimated to be about 1% for the IMI=1 and Vi% for the M=0 component. In Figure 4 the differences and similarities between the M=0 and IMI=1 avoided crossings are displayed. The 3j-symbols in the expression for the coupling -50-

cf *Pi and3F2 with intermediate D states are smaller for M=0 than for IMI=1 which is reflected in the narrower crossing gap Wo for M=0. Figures 4A1 and 4A2 illustrate the good agreement between the (noisy) experimental spectra and the fits obtained. The difference between the fitted curve and the experimental spectrum for the IMI=1 sublevel in Figure 4A2 (and Fig.5B) may be attributed to the non-linear response of the electron multiplier. Figures 4B1 and 4B2 give the positions of the two levels as a function of field strength, as in Figure 3.

A1 M = 0 F = 0.797 V/cm

z t \JA 0 200 40I 0 600 800 0 200 400 600 800 —• V(MHz) •V(MHz)

3 660 - I F2 >\ B1 360 - B2

13> _ 2 620 — . 320 —• W° t5 580 -~ 1 280 i i

3 1 I F> 1 F > 2 i / C1 2 C2 W i to UJ T t I'P,> 0.77 0.7J9 I0.81 0.83 0.87 AO.89 0.91 0.93 —• F(V/cm) —•F(Wcm)

3 l Figure 4: Theavoidedcrossing6s6tf F2 <->6s64p Pt/orM=0(1)andJM/=] (2) A: Experimental spectrum plus fitting curve B: Energy as a function of field strength C: Intensity as a function of field strength -51-

However, in Figure 4 level positions are given relative to the starting point of the laserscan to bring out the difference in shifts of the !Pi level (small polarizability) and the 3F2 level (large polarizability). Figures 4C1 and 4C2 display the intensities (arbitrary units) as a function of

field strength. It is clear that the curves are asymmetrical around Fc. This is due to the simul- 3 taneous excitation of both F2 and *Pi levels at zero-field, contrary to the case discussed in section 3.2. Although the intensity of the lP\ signal in zero-field is much weaker than the 3F2 signal its contribution could not be neglected. Especially the exact reproduction of the shape 1 of the interference-like profile turned out to be sensitive for the zero-field PJ signal strength. Furthermore it can be concluded that indeed the M=0 'Pj signal at zero-field must be weaker than the IMI=1 signal in order to obtain a good fit. A close inspection of Figures 4C1 and 4C2

reveals that the field strength for which both intensities are equal is larger than Fc in the M=0

case and smaller than Fc for IMI=1. As a consequence the interference-like profile for the M=0 case shows a reversed sequence of maximum and minimum intensity compared to the IMI=1 profile (e.g. see Figs.4Al and 4A2). It is obvious from Figure 4 that the interference- like profiles are observed only within a limited range of field strengths (range = 1/40 V/cm).

t

200 400 600 800 1000 200 400 600 800 • V( MHZ) —• V(MHz)

3 Figure 5: Experimental spectrum plus fitting curve for the avoided crossing F2 <-> "Pi for /M/=l at different field strengths. With the input parameters obtained from a fit of one spectrum it should be possible, by only changing the field strength, to reproduce other spectra. An example is given in Figure 5 where for F=0.891 V/cm (A) and F=0.905 V/cm (B) the experimental M=0 spectrum with the calculated curve, using the parameters deduced from the fit of the experimental spectrum at

F=0.900 V/cm (Fig.4A2), is shown. Again good agreement is obtained. The error of Wo in Table 1 is the standard deviation obtained from averaging several fits. -52-

4. Conclusions In this high resolution experiment two aspects of avoided crossings were studied. In the 3 1 simultaneous excitation of F2 and Pi Rydberg levels (n around 60) in one Iaserscan interference-like profiles were observed as the two states cross in an electric field. Due to field 3 inhomogeneities the F2 state, having a large polarizability, broadens rapidly with increasing field strength and at the crossing its width far exceeds the minimum crossing gap. Actually a summation of individual avoided crossings was observed. It was not feasible to reconstruct the experimental spectra using a diagonalisation procedure of the Stark energy matrix. Values of level energies and singlet-triplet mixing coefficients needed for such a calculation are not sufficiently accurate for n around 60. Instead the spectra were fitted to an intensity curve constructed by numerical integration of intensities near the avoided crossing over a field strength range determined by the field inhomogeneity. Input parameters were taken from zero-field data. The avoided crossings turned out to be extremely narrow; the minimum separation for IMI=1 was 33 MHz and 18 MHz for M=0. The common fraction in the wavefunctions of' Pi 3 and F2 producing the repulsion at the crossing is estimated to be about 1% for IMI=1 and l /2%forM=0. At higher field strengths also the 3Si Rydberg level was excited via admixture of odd 3 3 parity character in its wavefunction. The M=0 components of F2 and SX show the more regular, known type of avoided crossing; a gradual decrease respectively increase in intensity of 3 3 the F2 and Si excitations, until both have the same intensity at minimum separation. The 3 3 IMI=1 crossing could not be observed experimentally. In case of the F2 <—> Sj crossing 3 the minimum crossing gap exceeded the inhomogeneously broadened F2 linewidth. A cal- 3 3 culation of the avoided crossing F2 <-> Si using the diagonalisation procedure of the Stark matrix yields excellent agreement with the experimental value for the crossing gap. However, a discrepancy is found for the slope at which the levels approach. The experimental slope is smaller than the calculated one. The difference may be attributed to the limited number of levels used in the calculation. The introduction of more angular momentum levels does not solve the problem as level energies and singlet-triplet mixing coefficients are, as in the case of 3 the F2 <-> 'Pj crossing, not sufficiently well known.

References

1. Eit. Eliel: TTiesis Vrije Universiteit, Amsterdam 1982 2. ML. Zimmerman, M.G. Liltman, M.M. Kash and D. Kleppner: Phys. Rev. A 20 (1979) 2251 3. J.R. Rubbmatk, M.M. Kash, M.G. Littman and D. Kleppner: Phys. Rev. A 23 (1981) 3107 4. R.C. Stoneman and T.F. Gallagher: Phys. Rev. Lett. 55 (1985) 2567 and references therein -53-

5. W. Happer and R. Gupta: In: Progress in Atomic Speclroscopy. New York, Plenum Press 1978 6. TJ. Gallagher, L.M. Humphrey, W.E. Cooke, R.M. Hill and S.A. Edelstein: Phys. Rev. A16 (1977) 1098 7. T.H. Jeys, G.W. Foltz, K.A. Smith, EJ. Beiting, F.G. Kellert, F.B. Dunning and R.F. Stebbings: Phys. Rev. Leu. 44 (1980) 390 8. W. van de Water, D.R. Mariani and P.M. Koch: Phys. Rev. A 30 (1984) 2399 9. D. Kleppner, M.G. Littman and MX. Zimmerman: In: Rydberg states of atoms and molecules. Cambridge, Cambridge University Press 1983 10. C. Delsart, L. Cabaret, C. Blondel and R-J Champeau: J. Phys. B: At. Mol. Phys. 20 (1987) 4699 11. C.T.W. Lahaye, W. Hogervorst and W. Vassen: Z. Phys. D 7 (1987) 37 12. B.H. Post, W. Vassen and W. Hogervorst:/.Phys. B: At. Mo!. Phys. 19(1986)511 13. M. Aymar and P. Camus: Phys. Rev. A 28 (1983) 850 14. W. Vassen and W. Hogervorst: Z. Phys. D. 8 (1988) 149 15. W. Vassen, W. Hogervorst, T. Van der Veldt and C. Westra: submitted for publication 16. K.A.H. van Leeuwen: Thesis Vrije Universiteit, Amsterdam 1984 Chapter 5

Stark manifolds and electric field induced avoided level crossings in helium Rydberg states.

ABSTRACT

In a CW laser-atomic-beam experiment the linear Stark effect in the lsnp 1>3P Rydberg states of helium (n around 40) has been studied. The evolution of the angular momentum manifolds into the n-mixing regime was followed and avoided level crossings were observed. Stark manifolds were also calculated by diagonalisation of the complete energy matrix in the presence of an electric field. It turned out to be necessary to include up to five n-values in the calculations already at moderate values of the field to reproduce the data within the experimental accuracy (a few MHz), especially in the regime of the avoided crossings. -57-

1. Introduction In recent experiments in our laboratory fine and hyperfine structure in the lsnp Rydberg series of 3He and 4He up to high values of the principal quantum number n=80 as well as isotope shifts were measured [1,2]. This work is part of a program to study in detail the properties of bound Rydberg and autoionizing levels of two-electron atoms [2,3,4,5], where the electron-electron interaction manifests itself most clearly [6,7]. This program also includes an investigation of the influence of external fields, which may provide important additional information on the level structure of these atoms. E.g. measurement of electric dipole polarizabilities (quadratic Stark effect) not only yields information on energies and wavefunctions of the levels in question, but also, via the coupling by the electric field, on those of the nearby opposite-parity levels [8]. Also high angular momentum levels, not directly accessible from atomic ground states or low-lying metastable states may be populated in the presence of an electric field (linear Stark effect) [9]. The helium atom is the simplest, fundamentally most interesting two-electron system for which highly accurate calculations are available [10]. A review of calculations and experimental results on energies and fine-structure intervals of low-lying n-states (n<9) is given by Martin [11]. To further test the calculations there is a need for accurate data on high n-states, for which information is scarce [1], and for more accurate experimental results on low-lying states [11]. Information on the Stark effect in helium is also scarce. Values for the tensor U 1 polarizabilities of the lsnd D series up to n=7 [12,13,14] and for the Is4f F3 and Is5f ^3 levels [15] have been reported recently. This lack of accurate data on the bound lsn/ Rydberg levels of helium may be attributed to the experimental difficulty of populating these levels, which lie almost 200000 cm"1 above the Is2s ^0 ground state. The development of an effective source of metastable Is2s 'So, 3Sj atoms in conjunction with the production of tunable, narrow-band CW laser radiation in the wavelength region 260-335 nm in our laboratory created a facility to study in high resolution the lsnp 1>3P Rydberg series of3He and4He . As an extension of our previous work [1,2] here the first results on linear Stark effect studies in the lsnp 1>3P Rydberg levels of 4He with n around 40 are reported. The quantum defects of the *P and3P levels are quite small (S=-0.012 and 5=0.07 resp.), so they are nearly degenerate with higher orbital angular momentum levels. As the coupling energy of the excited electron with the electric field scales as n2 and the contribution of the quantum defect to the level energy as n~3 at relatively modest field strengths isolated (low /) levels may be considered degenerate with all higher angular momentum levels for high values of n and the linear Stark effect may easily be observed. Although the resemblance of helium and hydrogen Stark manifolds is striking the influence of levels having non-zero quantum defect is still present. In helium, contrary to the hydrogen case, non-crossing -58- phenomena [16] between levels of adjacent manifolds are readily observable. In section 2 of this paper the theoretical background of the linear Stark effect and calculational procedures will be presented. In section 3 the experimental setup is briefly described whereas in section 4 the experimental results are discussed and analysed. Conclusions are given in section 5.

2. Theoretical background The Hamiltonian describing the interaction of the atom with a uniform electric field F directed along the z-axis is:

HF=-PF=ezF (1) where P is the electric dipole operator. Stark spectra of helium Rydberg levels are calculated by diagonalising the complete energy matrix in a spherical base in the presence of an electric field following the method of Zimmerman [17]. The diagonal elements of this energy matrix are the zero-field energies whereas the off-diagonal elements contain the electric field contribution. The zero-field energies for helium, W^,/ , are calculated with the Rydberg formula with known quantum defects Sn>;: R

I is the ionization limit and R the Rydberg constant. In helium 1=198310.7723 cm l and R= 109722.2731 cm"1. Quantum defects for n around 40 and / up to 7 are given in Table 1. Table 1: Quantum defects of 4He 1 snl levels from Martin [11]

n 1 state 38 39 40 41 42 0 'S 0.139738 0.139737 0.139736 0.139735 0.139734 0 3S 0.296682 0.296680 0.296679 0.296678 0.296677 I 'P -0.012138 -0.012138 -0.012138 -0.032139 -0.012139 1 3P 0.068346 0.068347 0.068347 0.068348 0.068348 2 'D 0.002110 0.002110 0.002111 0.002111 0.002111 2 3D 0.002886 0.002886 0.002886 0.002887 0.002887 3 F 0.000438 0.000438 0.000438 0.000438 0.000438 4 G 0.000125 0.000125 0.000125 0.000125 0.000125 5 H 0.000048 0.000048 0.000048 0.000048 0.000048 6 I 0.000017 0.000017 0.000017 0.000017 0.000017 7 K 0.000008 0.000008 0.000008 0.000008 0.000008

These quantum defects were calculated with the explicit Rydberg/Ritz formula using the appropriate energy dependence derived from quantum defects up to n=8 [11]. Quantum defects for / > 7 are assumed to be zero. -59-

The fine structure in the 3P Rydberg levels of 4He for n around 40 is smaller than 1 MHz and consequently may be neglected. In this case a purely LS-coupled base may be used [18] and the off-diagonal matrix element representing the Stark effect can be expressed in such a base. A general matrix element may then be written as:

= } ^ (3)

for l'2=l2±l and zero otherwise. 1^ is the largest of I2 and Y2. Here it is assumed that the electric field only acts on the Rydberg electron. The transition integral RJ}'/< can be expressed in single-electron radial wave functions

Rn/(r) as follows:

0 To calculate this integral the method proposed by Zimmerman [17] using the Coulomb approximation of Bates and Damgaard was applied [19]. There is no selection rule for n in Eq.(3) so in principle all n are dipole coupled by the electric field. However the coupling between adjacent principal quantum numbers is relatively small compared to the dipole coupling within n due to smaller transition integrals (Eq.(4)). Consequently it will depend on the value of the field strength how many n-values must be taken into account to reproduce level positions accurately. E.g. to calculate the (anti-)crossing between two levels of adjacent n-manifolds at least these two n-values should be used in the calculation. To reproduce the relative intensities of the manifold peaks it is necessary to calculate the transition probability to excite a Stark state from a low lying state. The oscillator strength for a transition fron; a ground state W, L, ML to a Stark state W\ M' is [20]: 2

"W) (5) where U{JJ»/,» is the unitary transformation which diagonalises the Stark matrix. In the discussion of Stark manifolds it is more convenient to auapt quantum numbers used for the hydrogen case [Bethe] i.e. parabolic quantum numbers. Each component of the Stark manifold then corresponds to a Stark state lnkm> with k=ni-n2 in terms of the usual parabolic quantum numbers n; and n2. These quantum numbers are related by n=n1+n2+lml+l, so for fixed n and m, k ranges over the values [n-lml-l,n-lml-3 ,-n+!ml+l]. Furthermore -60-

the parabolic magnetic quantum number m is equal to ML- It appears as if the parabolic representation provides a better base for calculating Stark manifolds. However, in the parabolic base the zero-field states are not diagonal as in the spherical case. Also, because of deviations from the hydrogenic case (quantum defects) the Stark states lnkm> are not pure so the Stark effect is not diagonal in the parabolic base. Thus most of the advantage using the parabolic representation is lost. Since it is more convenient to calculate manifold intensity profiles originating from a low-/-valued non-degenerate level (Eq.(5)) in the spherical base this one is used in the diagonalisation procedure. As levels with different /-values are not degenerate at zero field interesting, non- hydrogenic phenomena may appear in the observation of manifolds. This paper focuses on avoided crossings between Stark states of different n due to higher order Stark effects. The repulsive interaction between these states originates from core interactions at zero-field which break the parabolic symmetry. These interactions in fact are represented by the quantum defects (Table 1). Komarov et al. [21] have derived an explicit analytical formula to estimate the minimum separation Wo between two Stark states from zero-field data: •w-1 f(n-l)/2 (n-l)/2 / 1 f(«'-l)/2 («'-l)/2 /] KWo-Om'jr** £ (2/+1) l+k)/2 {m_k)f2 _J lm+ky2 {m_kV2 I* (6) I7>\m\ I J I J The 3j-symbols relate the zero-field data in spherical base with the Stark levels represented in lae parabolic base.

3. Experimental setup A detailed description of the experimental setup (Fig.l) is presented elsewhere [1]. For this reason only a brief description, emphasizing details relevant for the present experiment, is given here. Metastable states of helium were produced by running a discharge between a skimmer and a tantalum needle inserted in a quartz tube. The helium gas flow expanded through the nozzle of this quartz tube via the skimmer into the interaction region. Metastable state production was estimated to be in the order of 1014 atoms per second per steradian with a ther- mal velocity of 2000 m/s [22]. Experimental data showed that in the interaction region the ratio of the population of Is2s 'So versus 3So metastable states is about 3. Doppler effects were reduced using a small nozzle orifice (0.2 mm diameter) and an adjustable collimating slit (0.3 mm, collimation ratio 1:600). The metastable beam was perpendicularly intersected with focused laserlight. The interaction took place between two capacitor plates shielded in a metal box. The separation between the plates was measured to be 5.80(5) mm. The residual Doppler width of a field free transition to a lsnp Rydberg level was about 10 MHz. The excited Rydberg atoms leaving the box were field ionized by applying a positive voltage of -61-

about 50 V to the box. The produced ions were accelerated into a quadrupole mass filter and the selected 4He ions counted with an electron multiplier. The signal was stored on a minicomputer also controlling the laser scan. Furthermore part of the (visible) output of the dye laser was sent through a length-stabilized etalon (fsr:148.9568 MHz). The Fabry-Perot signal was also stored on the minicomputer for calibration purposes.

RING DYE LASER

visible output

WAVELENGTH METER ELECTRONICS

150 MHz calibration interferometer

E.M.

IODINE STABI - USED ref out LOCK - IN HENE AMPLIFIER LASER in

COMPUTER

Figure 1: Schematic of the experimental setup: EM. electron multiplier; W.P. Wollaston prism; PD. photodiode. The ionization limit of 4He is at about 25 eV while the metastable states lie around 20 eV. High Rydberg states may be excited from these metastable states with UV laseriight. This was produced by intracavity frequency doubling a CW ring dye laser (Spectra Physics 380D) using temperature-tuned non-linear crystals [3]. To excite lsnp *P from Is2s 1So (n around 40) laseriight around 313 nm was produced using Rhodimine B dye and an RDP crystal. The transition Is2s 3So -»lsnp 3P is made with laseriight around 260 nm using the dye Couma- rine 6 and a KDP crystal. UV output was about 4 mW with 5 W pump power from an Ar+- laser. The dye lasers were frequency-stabilized yielding bandwidths in the order of 1 MHz in the ultraviolet region. UV scans up to 75 GHz were made. -62-

4. Results and discussion Manifolds originating from excitations of !P as well as 3P levels were studied. The only difference between singlet and triplet for high Rydberg states of helium is their quantum defect (see Table 1) as the fine structure splitting in the triplet states for high n may be neglected. So singlet and triplet manifolds are expected to show analogous behaviour as a function of the field strength. Since the singlet manifolds are more pronounced than triplet manifolds due to a higher population of the singlet metastable state in the discussion the singlet case will be emphasized. The polarization of the UV laserlight was parallel to the electric field axis (rc-excitation). Starting from an S-state together with the selection rule for an electric dipole transition (3j-symbol in Eq.(3)) only M=0 manifolds may be excited.

4 Figure 2: Stark manifold at F=10.02 Vlcm originating from the transition Is2s' So -> ls40p 'P in He.

4.1. "Isohted" manifolds An example of a singlet angular momentum manifold for n=40 an a field strength of 10.02 V/cm is shown in Figure 2. The total width of the recorded speeirum is 60 GHz. The distance between the outermost peak at the high frequency side of the n=40 manifold and the nearest peak of the n=41 manifold is 40 GHz, so the manifolds are still well separated. This manifold shows some characteristic features: strong signals at the low frequency side, a steep fall in : tensity with increasing frequency of the laser obviously resulting in suppression of manifold components and a reappearance of (weaker) signals at the high frequency side. This behaviour v typical for the M=0 manifolds of helium excited via singlet P-states and strongly deviates from the pattern observed in the manifolds originating from 6sng*G4 levels in barium [9]. The strong signal at the low frequency side in Figure 2 is connected to the excitation of the 40 'S-state (8=0.14, see Table 1). This state is directly dipole coupled to the P- state resulting in a large excitation probability. The singlet P-state with its negative quantum -63-

defect is originally located at the high frequency side of the manifold. Obviously the outermost levels "absorb" most of the oscillator strength at the cost of Stark states in the middle of the manifold. Experimentally Stark states with parabolic quantum number k between -39 and -25 and between —3 and 39 were observed in the case shown in Figure 2.

10 20 30 40 50 60 W(GHz)

Figure 3: Experimental {+) and calculated (•) relative intensities as a function of energy for the Stark manifold at F=10.02 V/cm as shown in Figure 2. To compare experimental results with theoretical calculations some data adaptation was necessary as not all manifold peaks had the same Doppler limited linewidth (10 MHz). Broadening due to field inhomogeneities was present. As in first order perturbation theory the energy of the hydrogenic Stark levels is proportional to 3/2 nkF [23] a field inhomogeneity F will mostly affect the outermost levels of the manifold with largest k-values. From the observed linewidth variation an inhomogeneity of 5 mV/cm was deduced. For this reason the peak intensity was taken to be the integrated signal strength concentrated in the centre of the resonance. The thus obtained results as derived from the spectrum shown in Figure 2 are given in Figure 3 where also the manifold pattern calculated with the diagonalisation procedure outlined in section 2 is included. To reproduce level positions to within the experimental accuracy at the MHz level it turned out to be necessary to include three n-values (n=39,40 and 41) in the diagonalisation. The overall agreement between experiment and calculation is good. For example the intensity minimum in the middle of the manifold is reproduced. From a comparison of calculated values and observations it follows that the levels with k in between -25 and -3 indeed were too weak to be observable. It appears as if the calculated intensities at the low frequency side are slightly underestimated and at the high frequency side overestimated compared to experiment. However, the calculation was based on constant laser power, which in fact was not the case for the large laserscans necessary to record the spectrr. -64-

Experimentally the laser power varied ir between 4.5 mW at the low frequency side and 3.5 mW at the high frequency siue in the case shown in Figures 2 and 3. Apart from the intensity distribution the triplet manifold shows the same behaviour as the singlet case. The intensity distribution in the triplet case is different as here the regular ordering of levels (3S-state below 3P-state below higher /-states see Table 1) holds.

F= 16.05 V/cm

5GHz

ILL I ILliJ Liu ill F = 14.40 V/cm

Id F= 12.75 V/cm t

F= 11.09 V/cm I 41 -40,0>

Figure 4: The n=40 (low frequency side) and n=41 (high fre- 4 40 39 0 quency side) manifold in He with increasing field llllLil.i' : ' ' LLkJ strength.

4.2. Avoided crossings

4.2.1. Avoided crossings of two levels of adjacent manifolds With increasing strength of the field the gap between adjacent manifolds diminishes as shown in the spectra of Figure 4 for n=40 and 41, again obtained in excitation from the singlet metastable state. Unlike the hydrogen case the levels do not cross with increasing field. An example is given in Figure 5 for the two outermost levels of the n=41 (I41,-4O,O>) and n=40 (I4O,39,O>) manifolds for electric fields between 16 and 16.5 V/cm. The origin of this typical -65-

INTENSITY t 5OO MHz

F= 16.44 V/cm

F=iaO9V/cm

Figure 5: Experimental mop ofthe avoided crossing ofthe AHe manifold levels /40,39,0> <-> /41,-40,0>. avoided crossing is the non-degeneracy at zero-field of levels with different values of /, contrary to the hydrogen case. Especially the S-states have significant quantum defects resulting in a coupling of e.g. the (n+l)s state both with the (n+l)p and np states in the presence of the field. In sum then there are dipole couplings between manifolds originating from different n and consequently Stark manifolds with different n-values cannot be treated as being independent as is the case of hydrogen. In Figure 6 the experimental level energy separation AW is plotted as a function of field strength in the region of the avoided crossing of Figure 5. Also included are the results of a calculation using the diagonalisation method. To obtain good agreement between experiment and calculations in the crossing region at least four n-values (n=39,40,41 and 42) had to be incorporated in the diagonalisation. In Table 2 experimental and calculated values for the minimum separation Wo in the avoided crossing I41,-4O,O> <-> I4O,39,O> both for the singlet and triplet manifolds are given. -66-

600 l4:,-4O,o> V /

500 /

400 / < o I I 0 i i 16.0 16.1 |1(52 16.3 16.4 •. -*F(Wcro) -400

-500 - / \ / 140,39,0 > \ -600

\

Figure 6: Avoided crossing /40,39.0> <-> )41,-40,0> in 4He. Dots: experimental. Curve: calculated 4 Table 2: Experimental and calculated values of the minimum separation Wo in the avoided crossing of the He manifold levels /4039,0> <-> /41,-40,0> (in MHz).

singlet triplet experimental 719.6 586.2 calculated 721.4 588.9 F (V/cm) 16.18 15.58

According to the previous discussion of avoided crossings Wo should be larger with increasing deviation from a hydrogen state and will be maximal when an isolated state is positioned halfway two manifolds, i.e. when its quantum defect is 0.5. From Table 1 it fol- 3 l 3 V lows that 8( S) = 25( S). However, W0( S) is not larger than WO( S) but even smaller. A possible reason is that the above given explanation concerns manifolds evolving from zero-field states with only one level deviating from the hydrogen case, whereas in helium not only S- states have a quantum defect but also P(and D,F,...) states. In addition in case of the avoided crossing in the triplet case the 41 3S state (I41,-4O,O>) has not yet completely merged with the n=41 manifold. So there is a larger separation between this state and the neighbouring 141,- 38,0> Stark state than between the other Stark states in this manifold. This will result in 3 a smaller energy gap Wo between this originally S state and the Stark state I4O,39,O>. -67-

Using relation (5) to calculate Wo a value of 573.6 MHz is obtained for the singlet case, 146 MHz below the experimental value. The difiference stems from the fact that in the model of Komarov et al. [21] only two adjacent n-values are considered, which is obviously not sufficient as follows from a comparison with the diagonalisation procedu e. For the triplet case relation (5) gives a value of 2424 MHz, which is even further off from the experimental value. As mentioned above this may be attributed to the fact that the triplet avoided crossing is not really between two manifold states.

INTENSITY t 1GHz

F= 34.76V/cm *fc>

V/cm

Figure 7: Experimental map of the threefold avoided crossing of the manifold levels /39,38,0> <-> 140,-3,O> <-> /41,-40,0> m4He. Also a value of the minimum energy separation between the Stark states I5O,-49,O> and I49,48,O> in the triplet case was determined (286 MHz). According to Zimmerman [17] the minimum crossing gap Wo roughly scales as vT4 as the density of Stark states for m=0 is proportional to n4. The ratio of the observed triplet gaps is 0.49, in reasonable agreement with the scaling law ratio of 0.45. -68-

Finally also the mixing of wavefunctions and the resulting relative peak intensities in the avoided crossing region v.ere calculated. In general the experimental intensities could be reproduced although the comparison was not straightforward due to the scatter in the observed signal strength.

4.2.2. Threefold avoided crossings In the preceding section the merging of two levels of adjacent manifolds was discussed. Also more complicated anti-crossing phenomena could be observed. An example is given in Figure 7 where at a field strength of about 34 V/cm the two outermost levels of the n=41 and 39 manifold (I41,-4O,O> and I39,38,O> respectively) and the I4O,-3,O> level of the n=40 manifold show a complicated avoided crossing pattern. When the even k states are far apart the level k=-3 of the n=40 manifold is not observable as it is located in the low intensity part (see Figs.2 and 3). With decreasing separation the I4O,-3,O> signal gains intensity at the cost of the outer two signals. For certain values of the field the outermost peaks completely vanish while the middle one has maximum intensity and shows a narrow line width (10 MHz).

2.O 141,-40,0 >

£ 1.5 O

I 10 L

05 - 140,-3.0 >

-0.5

-1.0

% -1.5

-20 139,38,0 >

Figure 8: Avoided crossing/39,38.0> <-> I4O,-3,O> <-> /41,-40,0> in*He Dots: experimental, Curve: calculated -69-

2.0 |41,-40,0>'

1.5

1.0

33 33.5 34 34.5 35 I -05 F(V/cm)

-1.0

-1.5 -

-2O - | 39, 38,0 >

Figure 9: Avoided crossing 139,38,0> <~> j41,~40.0> m4He Dots: experimental, Curve: calculated In Figure 8 the experimental level energy separations in the region of this crossing pattern are displayed as a function of F. Energy separations of the two outermost levels are plotted relative to the I4O,-3,O> level, which falls on the abscissa of Figure 8. In the plot four energy extrema W1-W4 in the crossing region may be recognized, values of which are collected in Table 3. To reproduce the data shown in Figure 8 and Table 3 at these relatively high values of the field strength a diagonalisation of an even larger energy matrix including five n- values (n=38,39,40,41 and 42) was necessary. The results of this calculation are included in Table 3 and Figure 8. In Figure 9 the energy separation of the two outermost levels alone is plotted, showing the regular avoided crossing pattern discussed in section 4.2.1 with a 1 minimum separation Wo = 1W21 +1W31 = 1587 MHz (experimental value) or 588 (calculated value). Figures 8 and 9 show that, starting at low F (30 V/cm), with increasing field strength the separation between the levels I41,-4O,O> and I39,38,O> decreases as if the level I4O,-3,O> is not present, up till some value (F around 34 V/cm) where its influence becomes apparent. Here the I41,-4O,O> level is repelled by the I4O,-3,O> level. This repulsion disappears again at F=34.3 V/cm where the repulsive interaction with the I39,38,O> level takes over. The limited -70-

influence of the I4O,-3,O> level may be explained considering the signal strengths. The intensity observed is a direct measure of the P-character mixed into the wavefunctions of the Stark states. It is now obvious that when the I41,-4O,O> and I39,38,O> levels are far apart (see Figs.7a and 7h) only their mutual interaction is important. As the I4O,-3,O> level in the middle has no excitation strength coupling with the outermost levels is not noticeable. With increasing F the I4O,-3,O> level gains signal strength up till some value (F=34.05 V/cm) where its intensity (I2 ) becomes the sum of the intensities of the outermost peaks (I2 = Ij+13 both theoretically and experimentally). The corresponding increase of P-character in the I4O,-3,O> wavefunction thus results in interaction with the outermost levels. At F=34.55 V/cm the intensity of the I4O,-3,O> excitation again is the sum of intensities of the other two peaks after a decrease in between. At still higher fields I4O,-3,O> vanishes from the spectrum and its influence again is negligible.

Table 3: Experimental and calculated values ofextrema W in the threefold avoided crossing of the 4He manifold levels 13938,0> <-> 140,-3,0> <-> J41 ,-40,0> (in MHz)

Wo w, w2 W3 W4 experimental 1587 383 -960 627 582 calculated 1588 385.5 -963 630 585

4.3. Remarks From the discussion of isolated angular momentum manifolds and avoided crossings in the n-mixing regime in sections 4.1 and 4.2 it follows that with increasing F more and more n-values must be included in the diagonalisation of the energy matrix to reproduce the highly accurate data. This puts high demands on the numerical methods and will ultimately limit the applicability of this diagonalisation procedure. In the case of the three-level avoided crossing discussed in section 4.2.2 e.g. five n-values had to be considered, resulting in the diagonalisation of a 200 x 200 energy matrix. At even higher field strengths other physical processes become important. In the presence of the field the ionization limit is lowered yielding an increased tunneling probability. The influence of the continuum is not accounted for using the diagonalisation method. In this case the alternative approach as proposed by Harmin [24] where the Stark effect is incorporated in the framework of quantum defect theory, may be applied. However, also in this theory large energy matrices have to be dealt with. Again the avoided crossings will be a sensitive tool to test the theory both for weak and strong electric fields. -71-

5. Conclusions The development of an efficient helium metastable beam source enabled the investigation of high lsnp Rydberg states with CW laser light in the presence of an electric field. As these P-states have almost zero quantum defect already at modest electric field strenghts transitions from quadratic to linear Stark effect (angular momentum manifold) may be observed. However, states with non-zero quantum defect (especially the S-state coupled to the P-state by the field) causes the atom not to act purely hydrogenic. In the electric field also states of different n are dipole coupled and n-mixing occurs. This gives rise to avoided crossings when different Stark states approach each other. These phenomena were studied for triplet as well as singlet states for n around 40 and m=0. As might be expected these avoided crossings are a sensitive probe for the mixing of different Stark states lnkm>. This also puts high demands on the calculational method. In this work the method of diagonalisation of the complete energy matrix was applied including up till five n-values. A straightforward approach of relating avoided crossings between two levels of adjacent manifolds with zero-field data failed, making clear that these phenomena in helium cannot be described accurately with a two level model. This was confirmed by the diagonalisation calculation where four n-values (n=39-42) were needed to reproduce the avoided crossing I41,-4O,O> <-> I4O,39,O> and five (n=38-42) to calculate the threefold avoided crossing I41,-4O,O> <-> I4O,-3,O> <-> I39,38,O> accurately within the experimental error (3 MHz). However, for higher field strengths this method will not hold because of i.a. the influence of the continuum. An alternative approach is QDT incor- porating the Stark effect. Also for this theory the avoided crossings will be a sensitive test. This will be a subject of future study in our laboratory.

References 1. W. Vassen and W. Hogervorst submitted to Phys. Rev. A 2. W. Vassen: Thesis Vrije Universiteit, Amsterdam 1988 3. E. Eliel: Thesis Vrije Universiteit, Amsterdam 1982 4. B.H. Post Thesis Vrije Universiteit, Amsterdam 1985 5. E.A.J.M. Bente and W. Hogervorsu Phys. Rev. A 36 (1987) 4081 6. U. Fano and A.R.P. Rau: Atomic collisions and spectra. Academic Press 1986 7. M. Aymar: Phys. Rep. 110 (1984) 163 8. K.A.H. van Leeuwen: Thesis Vrije Universiteit, Amsterdam 1984 9. C.T.W. Lahaye, W. Hogervorst and W. Vassen: Z. Phys. D 7 (1987) 37 10. G.W.F. Drake: Nucl. Instr. and Meth. in Phys. Res. B 31 (1988) 7 11. W.C. Martin: Phys. Rev. A 36 (1987) 3575 12. W. Schilling, Y. Kriescher, A.S. Aynacioglu and G. von Oppen: Phys. Rev. Lett. 59 (1987) 870 -72-

13. G.G. Tepehan, H-J. Beyer and H. Kleinpoppen: J. Pbys. B: At. Mol. Phys. 18 (1985) 1125 14. W-D. Perschmann, G. von Oppen and D. Szostak: Z. Phys. A 311 (1983) 49 15. A.S. Aynacioglu, G. von Oppen, W-D. Perschmann and D. Szoslak: Z. Phys. A 303 (1981) 97 16. J.R. Rubbmark, M.M. Kash, M.G. Littman and D. Kleppner: Phys. Rev. A 23 (1981) 3107 17. MX. Zimmerman, M.G. LiUman, M.M. Kash and D. Kleppner: Phys. Rev. A 20 (1979) 2251 18. E.S. Chang: Phys. Rev. A 35 (1987) 2777 19. D.R. Bates and A. Damgaard: Phil. Trans. R. Soc. 242 (1949) 101 20. R.D. Cowan: The theory of atomic structure and spectra. Berkeley, Los Angeles, London: University of California Press 1981 21. I.V. Komarov, TJ>. Gfozdanov and R.K. Janev: J. Phys. B: At Mol. Phys. 13 (1980) L573 22. D.W. Fahey, W.F. Parks and L.D. S,chearer: J. Phys. E 13 (1980) 381 23. H.A. Bethe and E.E. Salpeter: Quantum mechanics of one- and two- electron atoms. Berlin, Gottingen, Heidelberg: Springer 1957 24. D.A. rfexmta: Phys. Rev. A 26 (1982) 2656 Chapter 6

Electric field effects in weakly autoionizing 5dnf J=5 states of barium

ABSTRACT

In a CW laser-atomic-beam experiment the Stark effect in weakly autoionizing 5d3/2nf J=5 Rydberg states of barium (n around 20,34,60 and 70) was investigated. Stark spectra at n=60 were calculated using Multichannel Quantum Defect Theory extended tu include electric field effects. The gross features of the complicated experimental spectra could be understood in this MQDT framework. However, detailed comparisons were not possible as sufficiently accurate information on 5dnl autoionizing series coupled in the field to the 5dnf series is not available. Also the method of diagonalisation of the Stark matrix, commonly only applied for bound states, was used and results compared with the MQDT calculations. As expected the diagonalisation procedure only reproduced field-dependent level energies in those cases where the coupling of the states involved with the continuum is almost negligible. -75-

1. Introduction In the past decade extensive laser spectroscopic studies have been performed on the influence of electric fields on bound Rydberg states of alkali and alkaline-earth atoms [e.g. 1,2,3,4,5]. Detailed information both on quadratic and linear Stark effects has been collected. The quadratic effect normally occurs in non-hydrogenlike excited states of atoms with low values of the orbital angular momentum (/), i.e. in states with large quantum defects. States with high values of / and principal quantum number n are nearly hydrogenlike (quantum defects almost zero) and consequently the linear Stark effect is observable. Angular momentum manifolds fanning out (approximately) proportional to the strength of the field can be excited. There is a growing interest for studies of the influence of electric fields on autoionizing states where the continuum plays an important role [4], Drastic changes in shapes and widths of autoionizing resonances have been observed in experiments with pulsed dye lasers as well as electric-field-induced interferences [6,7,8,9,10,11,12,13]. These phenomena are manifestations of the complicated interplay between continuum and discrete channels in the presence of the field. The reverse process is dielectronic recombination, which is dominantly present in high-temperature plasmas. The microscopic field in such a plasma will influence the recombination rates and consequently its properties [14]. Inclusion of Stark effect data in calculations can result in improved understanding of plasmas. Autoionizing Stark spectra were calculated using e.g. the diagonalisation procedure of the energy matrix in the presence of the field [15] or the formalism developed by Fano [16] for a many-discrete-level-single-continuum configuration interaction [17]. However, these methods yield more or less satisfactory results only for special cases. A better calculation^ approach of the Stark effect was suggested by Harmin [18,19,20]. He incorporated effects of external fields in Quantum Defect Theory (QDT). As in his theory only single channels were considered it was not straightforward to apply it for autoionizing states. It can in principle only be applied in cases where the influence of the continuum on (single) channels is not intrinsic but caused by the field-induced lowering of the ionization limit [21,22,23]. Exam- ples of this type of calculations are discussed e.g. by Liu et al. [24] for sodium (one-electron system) and by Van de Water et al. [25] for helium (two-electron system). Sakimoto [26] extended Harmin's theory to the Multichannel case. The outline of the theory for dielectronic recombination is discussed in a paper of 1987 [27]. In this paper we present first results of a CW laserspectroscopic investigation of the Stark effect in weakly autoionizing states of barium as well as first results of calculations based upon the MQDT method. In our high resolution experiment salient details of autoionizing Stark spectra were revealed. These experimental results may be used for a stringent test of the calculational procedure. The calculated results and experimental data are compared and -76- discusscd in section 4. In section 2 a summary of Sakimoto's theoretical approach is given. Section 3 deals with the experimental setup and contains the results of the Stark effect investi- gations.

2. Theory The most elaborate non-perturbative theory on the Stark effect for Rydberg states is MQDT adapted for the electric field case. Here the theory as presented by Sakimoto [26,27,28] is summarized following the lines of conventional MQDT as formulated by Seaton [29]. Starting point for the quantum defect treatment is the distinction between short- and long-range effects. The short-range effects are limited to a core region (r

g and g are the regular and irregular Coulomb functions respectively, W is the energy and m/ is the z-component of the orbital angular momentum / [29]. 'i>.-~ "ore potential, though complicated, also has spherical symmetry. In hydrogen the electronic ..wtion is represented by Eq.(la). However, the presence of a non-hydrogenic core modifies the total wavefunction G° of the electronic motion. In MQDT this solution G° can be written as (in matrix notation) [29]:

0 (2) RyT/nVYfoii is the so-called reactance matrix, y denotes the state of the core. In the simplest general case (bound Rydberg series without perturbers) the r°-matrix contains the phase shift of the hydrogenic wavefunction, dependent on the quantum defect ji. To discuss autoionization (or dielectronic recombination) it is more convenient to consider the problem in the scattering formalism by introducing the functions -77-

(3a)

V (3b) \|f+ and \j/2 represent outgoing and incoming waves, respectively for open channels. The wavefunction G° can be written as: 0 G°=Y«-v«Z (4) X° is related to R° by X° = (l+JRO)(1-/R°r1 (5) The scattering matrix can explicitly be expressed in terms of X and v [29]:

S°=Xo°o-zSc [xSo-e^'xl, (6)

Here o and c stand for open and closed respectively and v=n-n is the effective quantum number [29]. In principle the S°-matrix contains all information to calculate autoionization spectra in zero-field. When an electric field F is switched on only the long range potential is influenced: V(r)=-l/r+Fz (7) Here the direction of the field is chosen along the z axis. Potential (7) has a saddle point (j) at

z=-rs=-l/YF (a.u.) for which V=WS=-2W (a.u.), the so-called classical ionization limit [30]. Electrons with an energy W>W^ can easily escape and are free. The Stark effect of the

Rydberg atom in a resonance state (W) by: ) or % Ti=r(l-cos9) or r\ = r-z (8)

Two independent solutions of the separated Schrodinger equation are (r>r0) [26]: »• (9a) mt> (9b) The solutions (9) depend on three constants of motion, i.e. the energy W, the separation constant P and the parabolic magnetic quantum number m [30]. As the ^-motion is bound this constant P takes discrete values at a given energy W [31]:

(10) -78-

where n\ is a non-negative integer. The T|-motion is quasi-bound [31] and yields a quantum number n2. The parabolic quantum numbers are related to the principal quantum number n by [30]:

n=ni+n2+\m\+l (11) The energy of a Stark state in a first order approximation is (in a.u.):

Here k is the parabolic quantum number: k=ni — n2, see Eq.(l 1). For fixed n and m, k ranges over the values [n-lml-l,n-lml-3, ,-n+lml+l]. In the transition from spherical to parabolic coordinates m/=m. The electronic wavefunction in the presence of an electric field may be constructed in analogy to Eq.(2): * G=Y+?R (13) To calculate Stark spectra from zero-field analyses the reactance matrix R (Eq.(13)) should be connected with R° (Fq.(2)). From inspection of potential (7) it follows that a large region exists where the Stark contribution is negligible compared to the Coulomb term [19]:

ro<:r*:rs (14) In this region (14) solutions for the electronic motion can equally well be expressed in terms of the parabolic Coulomb functions (Eq.(9)) as in terms of the spherical Coulomb functions (Eq.(l)). Now the appropriate frame transformation Up/ between the two different solutions of the Schrodinger equation, connecting the Stark case with zero-field input parameters [26] can be constructed. The matrix R/a'm'.yamm Eq.(13) can then be obtained by transformation of

the R?rOT.,7fel,-matrix [26]:

R = UR°U' (15) Here the matrix U has the elements:

fCYySn8OT'imi W>WS

's

Again it is more convenient to employ the S-matrix formalism. Analogous to the transition from R° to S° in Eq.(5) it follows that (17) and the scattering matrix [27,28,29]:

Zco (18) -79-

Here A is a phase integral [27]. The oscillator strength for bound (a) -» free (j) transitions is given by [32]:

2 -j-fja{E)=ll3{E-Ea) |<¥,(E)ldl«Fa>| (19)

with E the total energy of the system E = Ey+W (Ey being the energy of the core). The total wavef unction ¥j(E) of the system describes the actual motion of the excited electron and the residual ion core. This wavefunction can, with the help of the scattering matrix (18), be expressed in the zero-field Coulomb functions [28]. Next with Eq.(19) Stark spectra can be calculated. Input parameters for the program to calculate Stark spectra with the MQDT method are: the zero-field R°-matrices for all dipole-coupled states, the value of the electric field F, the magnetic quantum number m, the angular momentum of the excited zero-field level / and the proper ionization limit I.

3. Experimental setup and results

3.1. Experimental setup With the localisation of the metastable 5d2X Q$ level in barium, single-photon excitations to 6snh bound Rydberg and 5dnf autoionizing states became possible [33,34]. 5dnf series were excited with light from a frequency-stabilized Stilbene 3 ring dye laser (Spectra Physics 38OD) and showed extremely narrow autoionization linewidths. The laserlight perpendicularly intersected a collitnated atomic beam to reduce Doppler effects. The residual Doppler linewidth for the zero-field case was about 10 MHz. The metastable state was populated by collisions in a cloud of slow electrons around the heating filament in front of the beam-producing oven [33]. The interaction region was situated between two capacitor plates (separation about 6 mm). The electrons produced in the autoionization process were accelerated towards one of the plates, containing a fine wire mesh, and collected on a channeltron electron multiplier. The signal was stored on a PDP-11 computer also controlling the laserscan. The spectra were calibrated with a 750 MHz e'clon whose transmission peaks are also stored on the computer. In autoionizing spectra many interesting phenomena due to the field-induced dipole coupling of the excited state with adjacent more strongly (or weakly) autoionizing states can be observed. Spectra were taken for different values of n (n=20,34,60 and 70). For each principal quantum number n the observed pattern repeats itself provided series perturbations are absent. Utilizing the maximum scan range of the laser (up to 30 GHz in one laserscan) Stark spectra originating from n=60 and n=61 could be recorded simultaneously in one scan. The following discussion will focus on the n=60,61 spectra. -80-

5GHZ

t

F= 0.42

F= Q2

F=0.0

Figure 1: The autoionizing n=60 Stark manifold in barium originating pom the zero-field transition 2 5d ' G4 -» 5d3/260f/

3.2. Experimental results

In Figure 1 spectra of the 5d3/260f autoionizing states in an electric field are shown. In zero-field one J=5 and two J=4 5dnf states are observed. Due to the lower excitation probability of the J=4 states the recorded Stark spectra dominantly originate from the J=5 state. There- fore the discussion focuses only on the J=5 series. When the influence of the electric field is considered in a perturbational treatment no first order but only second order (and higher) contributions to the energy for isolated levels occur. This was indeed observed for low values of n (n=20) where the splitting according to the quadratic effect gave rise to the excitation of six resolved IMj I components for the J=5 state. However, the deviation of the 5d60f J=5 level from the (degenerate) hydrogenic level is small (quantum defect \L only 0.074). Approximat- ing both the electric field and the quantum defect as a perturbation to the hydrogenic level it follows that the electric field contribution to the energy will exceed the quantum defect term when 3/2Fn2 >nn~3 [3]. For such field strengths the level then can be assumed to be degenerate with all higher angular momentum levels and linear Stark effects are observed. Then J is no longer a good quantum number. Only m/(=m) of the excited electron is conserved, resulting in a splitting of manifold levels in !ml components due to higher order effects. For Stark -81- states originating from 5dnf levels Iml can have the values 0,1,2 and 3. The splitting according to Iml observed in e.g. the n=34 manifold. As is evident from Figure 1 the linear Stark effect already occurs at weak fields for the highly excited 5dnf states. At F=0.42 V/cm a quadratic Stark effect (splitting of IMI components) still is observable for the J=5 level, but the onset of the manifold is already visible. Compared to the zero-field excitation the 5dnf levels in the field (F=0.42 V/cm, Fig.l) are broadened and positioned on a broad resonance, together with the manifold pattern. This broadening has two causes. (1) As high Rydberg states are extremely sensitive to electric fields even small inhomogeneities strongly L.3uence the linewidth [35]. In an earlier experiment [3] a field inhomogeneity of about 10 mV/cm was deduced for the capacitor used in the experiment (2) Due to the coupling with the continuum states above the first ionization limit will autoionize. The linewidth of such a state depends on the strength of the configuration interaction between the discrete state and its continuum channels. As the coupling between the 5d3/2nf series and its continuum 6si/2eh channels is weak [34] extremely narrow linewidths were observed. For n=60 in fact the Doppler limited linewidth of about 10 MHz FWHM (see Fig.l) is registered. However, the electric field mixes states with opposite parity. 5dn/ states with / > 3 will have even smaller linewidths than 5dnf, but the 5dnd, 5dnp and 5dns states have high autoionization rates. The influence of the corresponding broad resonances is evident in Figure 1; especially the 5d64s J=2 resonance with quantum defect jx = 4.154 [36] is responsible for the characteristic behaviour at the onset of the n=60 manifold as it is located close to the 5d60f J=5 state. The broad feature in combination with broadening of the 5d60f states (apart from the additional line broadening due to field inhomogeneities mentioned earlier), is due to the enhanced coupling with the continuum through this 5d64s state. The interference pattern of the manifold is real. In fact it consists of a series of separate Fano profiles [16] generated in the competition in the excitation of discrti-- manifold Stark states and the continuum.

A further increase in field strength (Fig.l, F=1.25 V/cm) produces a broad continuum background interacting with the n=60 manifold. The broadening of Stark states at the wings of the manifold is mainly due to field inhomogeneities as these Stark states have the largest field induced dipole moment (largest k, see Eq.(12)) and consequently are most sensitive to electric fields [37]. The polarization of the laserlight was in principle perpendicular to the electric field axis (c-excitation). This yields for the trans:

region. So it is also of interest to study the region in between two manifolds. Figure 2 displays an experimental spectruni with a laserscan over the n=60 to the n=61 region. Striking features are the broad resonance at W=15 GHz (labelled 4) and the narrow peaks around 13 GHz (labelled 1,2 and 3).

5 10 15 20 25 30 W(GHz)

Figure 2: Experimental spectrum of the energy region between n=60 (zero-field position 0 GHz) and n=61 (zero-field position 29.6 GHz) for F=2 Vlcm. For peaks identified 1-4 see text section 4.2.

>- — t/i i

z±1 — I t IhJUlL, • Li O 5 10 15 20 25 30 h W(GHz)

Figure 3: Calculated spectrum of the energy region between n=60 (0 GHz) and n=61 (29.6 GHz) for F=2.000 Vlcm and m=0. -83-

4. Calculational results and discussion

4.1. Comparison MQDT method with experimental results

4.1.1. Input for MQDT program The primary goal of this study is to qualitatively explain the autoionizing phenomena observed in the presence of an electric field. The calculation and discussion emphasize the case for F=2 V/cm as shown in Figure 2. A theoretical approach which includes the coupling of bound channels with the continuum is MQDT of the Stark effect, briefly outlined in section 2. To calculate Stark spectra in this MQDT framework detailed knowledge on the zero-field properties of all (autoionizing) series coupled by the field is important. To simplify the zero-field analysis only states far away from perturbers are considered (as in the experiments) so that the problem effectively reduces to a two-channel case, i.e. the discrete channel under study (indicated with index 2) and its continuum (indicated with index 1). The R°-matrix (Eq.(2)) than only contains three independent elements: RJI, R?2 and R;j2- Giusti-Suzor and Fano [38] give relations to calculate Ry from linewidth and quantum defects, quantities that can be determined experimentally:

(20) COSJI|A #22 =tanjrp. Tv3 is the reduced (half) width of the unperturbed Rydberg series (in a.u.). A computer program based on the theoretical concepts outlined in section 2 was made available by Sakimoto [28]. We have adapted it for the case of the 5dnf autoionizing states in barium. Apart from the R°-matrix elements (Eq.(20)) the field strength F, magnetic quantum number m and the energy region of interest are input parameters. Both the quantum defect and linewidth steeply decrease with increasing / so that the corresponding R°-matrix will converge to zero. For the 5d60f state the quantum defect \i has the (small) value 0.074 and a linewidth T < 7 MHz. For these reasons states with / > 3 are considered liydrogenic with Ry=0. Possible excitations from the 5d2 *G4 metastable state include both the 5dnf and 5dnp autoionizing Rydberg series. As 5dnp states were not observed in a zero-field experiment only direct excitation of the 5dnf states is taken into account. The R° parameters for the 5dnf series were calculated from zero-field MQDT parameters [39]. In the field the coupling with the autoionizing states 5dnd J=4,5dnp J=3 and 5dns J=2 has to be considered. As no information on the 5dns J=2 states was available from literature an additional experiment was performed [36]. In a two-step experiment with pulsed dye lasers the 5dns and 5dnd autoionizing Rydberg -84-

21 series were excited from the 6s S0 ground state via the intermediate 6s6p *Pi level. Due to the relatively narrow linewidths of high-n autoionizing levels with high total angular momentum the relevant information was deduced from lower-n Rydberg states (n around 20) far away from perturbers. Accurate values for the R°-matrix elements of the 5dnd J=4 series could be extracted from the MQDT analysis published by Aymar [40]. These values were confirmed by the results of our pulsed experiment on the 5dnd states. The largest problem gave the 5dnp J=3 autoionizing series. Only one value of the quantum defect is known from literature [41]. The linewidth was assumed to have a value in between the values for the 5dns and 5dnd series. The Rfj values of the relevant autoionizing series used in the calculations are given in Table 1.

Tablel: Values of the R%-matrix elements (Eq.(20))for the Sdns, Sdnp, 5dnd and 5 dnf autoionizing series.

pO pO

5dnsJ=2 0 0.317 0.525 5dnpJ=3 0 0.442 -1.289 5dndJ=4 0 0.460 -6.314 5dnfJ=5 0 0.015 0.237

4.1.2. Comparison between experiment and calculated m=0 spectrum The 5dns Rydberg series play a role only for m=0 so the most significant differences from a pure hydrogenic manifold behaviour will be observed in this case. In Figure 3 the calculated m=0 spectrum is shown for F=2.000V/cm. This spectrum has the same horizontal energy scale as the experimental spectrum of Figure 2. Starting point (0 GHz) is the zero-field n=60 hydrogenic level. The zero-field n=61 hydrogenic level is than at 29.6 GHz. For the comparison it has to be noted that due to field inhomogeneities narrow Stark states broaden relatively more than already broad Stark features. It appears that the m=0 calculation is capa- ble of reproducing some of the dominant features in the experimental spectra. The broad resonance around 15 GHz in Figure 3 is a remnant of the m=0 component of the 5d63d state. This peak can be recognized in Figure 2 as the broad resonance labelled with 4. Remarkable are the local maximum around 10 GHz and minimum around 3 GHz in the intensity envelope of the n=60 manifold in Figure 3. It can, although less pronounced, be retraced in Figure 2. The maximum seems to be shifted towards lower frequencies. This may be an artifact in the experimental observation; field inhomogeneities have a larger influence on the narrow Stark states around 10 GHz than on the levels around 8 GHz (Fig.3) resulting in a relatively stronger decrease in intensity. The intensity minimum around 3 GHz (Fig.3) is due to the presence of the 5d64p state which in zero-field is assumed to be located at this position. The -85-

structure is in fact a Fano-interference profile created by the interaction between the discrete manifold states and the 5d64p level which is strongly coupled to the continuum [17,42]. Similar profiles at approximately the same frequencies are also found in spectra calculated for the lml=l component. So the experimentally observed local maximum and minimum are a summation of m=0 and lml=l Stark spectra which also can affect the positions of the observed local extrema compared to calculated values. In Figure 3 another Faro-interference profile is present in the n=61 manifold around 29 GHz. This profile results from the interaction between manifold Stark states and the nearby 5d6Ss state and is of course only visible in calculations for m=0. It cannot be found in the experimental spectra (Fig.2) as it is obviously obscured by lml=l,2 and 3 components. The narrow structures around 13 GHz (labelled 1,2 and 3 in Fig.2) are not present in the m=0 calculation as they stem from lml=2 excitations (see next).

3.0 a)

20 -

1.0 -

o 3.0 b)

20 -

10 -

o t 30 c) 20 -

10 -

12.0 I14.0 16.0 18.0 20.0 • W(GHz)

Figure 4: Calculated spectra for the intermediate region between n=60 and n=61 manifolds for F=2.000 Vlcm. Intensity in arbitrary units, a: m=0, b: /mf=l, c:/m/=2 -86-

4.1.3. Spectra calculated for |m|=0,l and 2 components The calculation of spectra with the MQDT program is extremely time-consuming. For this reason intensity spectra for different Iml components have been calculated only over a limited frequency range. The interval from 11 to about 19 GHz was selected as in this region of the spectrum interesting phenomena were observed. The spectrum for the lml=3 component is not further considered as the levels with /<3, which are responsible for the observed structure have no lml=3 component. The lml=3 manifold will be, apart from the small influence of die Sdnf levels, almost purely hydrogenic. In Figure 4 remarkable differences in the calculated spectra for various Iml components are shown. In the m=0 spectrum (Fig.4a) the remnant of the 5d63d state is located halfway in between the n=60 and n=61 manifold (at IS GHz), mainly due to the opposing influences of the Sd64p and 5d65s state. As the Sdns states have no lml=l component a shift of the Sdnd lml=l sublevels towards higher frequencies occurs. This is evident in Figure 4b where the 5d63d sublevel has already merged with the Gow frequency side of) the n=61 manifold. This results in broader resonances (T~ 100 MHz) compared with the linewidths at the high frequency side of the n=60 manifold (r= 1 MHz) which are not affected. The spectrum for the lml=2 component (Fig.4c) depicts quite a different behaviour. At the right side of the figure (16-19 GHz) the low frequency wing of the n=61 manifold is observed. These levels have, as die Stark levels at the high frequency side of the n=60 manifold (around 11.5 GHz), extremely narrow linewidths (F< 1 MHz). This implies that the influence of the continuum is weak. As only the Sdnd lml=2 sublevel has a pronounced coupling with the continuum, the influence of this level at a field strength of 2 V/cm is obviously limited to a narrow energy range, or, to few manifold levels only. The levels of the n=61 manifold displayed have a large intensity compared to the n=60 manifold states (about a factor 20). A reason for this intensity difference between low and high frequency manifold peaks is that the zero-field excited Sdnf states ate located at lower frequencies than the corresponding degenerate hydrogenic states (}i > 0). This yields a higher excitation probability for the n=61 manifold levels compared to the Stark states of n=60 in Figure 4c. Note that there are differences in the intensity scales of Figures 4a,b and 4c, indicating that mainly the lml=2 spectrum is observable experimentally. However, the inhomogeneous broadening by the field of these narrow lml=2 Stark states suppresses this dominant excitation pattern (compare Figs.4c and 2).

The most intriguing feature in Figure 4c is the peculiar structure around 13 GHz. It strongly resembles the structure at the same location in Figure 2. What is observed here seems to be the interaction of a n=60 manifold level with the Sd63d lml=2 sublevel yielding an avoided crossing [35,43]. -87-

4.1.4. Avoided crossings between the 5d63d level and n=60 manifold components In Figure 5 the calculated anti-crossing phenomenon as a function of field strength is shown on an enlarged scale. Up till about F=1.95 V/cm the 5d63d lml=2 level hardly shifts and only gains intensity. The Stark levels at the high frequency side of the n=60 manifold on the contrary shifts rapidly towards the 5d63d level. From F=1.990 V/cm to F=2.030 V/cm the avoided crossing occurs between the outermost n=60 manifold level (labelled in parabolic notation as I6O,57,2>) and the 5d63d level. Increasing the field strength even more also the I6O,55,2> state undergoes an avoided crossing with the 5d63d level. The linewidth of the I6O,57,2> state changes considerably in the crossing region but finally reduces again to its value before the crossing (T < 1 MHz). Its intensity however, keeps on growing (see spectrum for F=2.075 V/cm). It is obvious that these avoided crossings are highly sensitive for electric fields; a crossing occurs within a change of 0.05 V/cm.

F= 2.075 V/cm

I 60,57,2 >

1/7 hi 160,53.2 > F= 2.010

t F= 1.990

^160.55,2 >/ F= 1.950

F= 1.900

^ 60,57, 2 > F=?8OO

1 5d63d> F= 1.600 125 1S0 13.5 W(GHz)

Figure 5: Calculated spectra fonthe avoided crossing between the Sd63d level and outermost states of the n=60 manifold for /m/=2 as a function of field strength. Stark states are identified with their parabolic quantum numbers (njkjml. see Eq.(ll)). Intensity in arbitrary units. -88-

The experimentally observed structure in Figure 2 around 13 GHz shows a stage of the avoided crossing of a n=60 lml=2 manifold state and the 5d63d level. However, best agreement between observed and calculated spectrum is not obtained for F=2.000 V/cm but for F=2.075 V/cm. Such an error in the experimental determination of the field strength is possible because of a relatively large and varying offset voltage needed in the used capacitor system. Together with the field uncertainty, the error in the determination of the field strength was about 0.1 V/cm. So the best matching field strength falls within the experimental error. In fact the large sensitivity of the different stages of the avoided crossing for the precise value of the field strength offers an alternative way for an accurate determination of the field. The (small error) for the actual field strength does not influence the general picture outlined for the m=0 and lml=l components. In the comparison of the experimental spectrum (Fig.2) with the calculated picture (Fig.5, F=2.075 V/cm) again it has to be taken into account that experimentally observed peaks show an inhomogeneous broadening. The experimental energy separations AW between the outermost peaks and the middle one (labelled 1,3 and 2 in Fig.2) are about

AWi_2 = 170 MHz and AW2_3 = 660 MHz, whereas the calculated values are AW!_2 = 177

MHz and AW2-3 =410 MHz respectively. Although the agreement for AWt_2 is reasonable there is a large discrepancy for AW2-3. This discrepancy may be attributed to inaccurate input data for the calculation. A small variation of the quantum defect of the 5d63d level has great influence. Furthermore, quantum defects for states with / > 3 are assumed to be zero. This also can have effects on the positions of states which have to be calculated at MHz accuracy. This requires extremely accurate values for the R°-matrix elements of the different states involved. For the present experiment the values of the R°-matrix elements are not sufficiently well-known (especially for the 5dns and 5dnp states) and only conclusions concerning the overall trend can be drawn.

4.2. MQDT method versus diagonalisation procedure In Figure 6a the energy positions of states around 13 GHz as calculated with the MQDT program for the lml=2 case are given as a function of field strength. A similar picture was also constructed using the procedure of diagonalisation of the complete energy matrix [1,3] (Fig.6b). Four n-values (n=59,60,61 and 62) were included in the calculation [43]. The excellent agreement between Figures 6a and 6b is remarkable although the procedures of calculating energy positions are essentially different. For example the minimum separation Wo in the avoided crossing between the 5d63d level and I6O,57,2> (indicated in Fig.6) obtained with the

MQDT method is Wo=I18 MHz and for the diagonalisation method Wo=117 MHz. The reason for this similarity is that, although only MQDT incorporates coupling with the continuum, this coupling is almost negligible in the lml=2 case. For lml=2 only the 5dnd level has -89-

some interaction, but even for this state it is relatively weak compared to the interaction of 1=0 and / =1 states. For example the linewidth of the 5d63d level is only 70 MHz (see Fig.5) corresponding with a product Fv3 =0.0033 a.u. So in approximation this coupling with the continuum may be neglected for the calculation of energy positions. However, to calculate lineshapes of transitions (Fig.S) the MQDT method has to be applied. The influence of the continuum could not be neglected for the 5dns and 5dnp states, as was concluded from a calculation with the diagonalisation method for the m=0 and lml=l excitations. It failed to reproduce level positions, as calculated by the MQDT method, especially close to the broad autoionizing resonances.

140 r 14.0

- I5d63d>

13.0

l 60,45.2 > 12OU, I 12O 1.0 2.S. 1O

Figure 6: The energy of autoionizing /m/*2 Stark states relative to the zero-field n=60 level as a function of field

strength. Wo is the minimum separation of the first avoided crossing between the 5d63d level and the n=(S0 manifold states. Stark states are identified with their parabolic quantum numbers (njcjmj, see Eq.(ll)). a: Calculated with the MQPT method, b: Calculated with the diagonalisation method Figure 6 shows the avoided crossing of the 5d63d level with succeeding levels of the n=60 lml=2 manifold. The avoided crossing with the I6O,57,2> is most pronounced (sharpest). With increasing field strength the crossings gradually flatten out, indicating that the 5d63d level gradually merges into the manifold.

4.3. General remarks

The primary goal of this study was to test the MQDT version for the Stark effect. The qualitative behaviour of autoionizing states (in this case 5dnf) exposed to an electric field is reproduced well by MQDT. However, the calculations did not match the experimental results in details as e.g. was shown in the case of the avoided crossing between the 5d63d level and the n=60 manifold states. This may probably be due to insufficient knowledge of zero-field input parameters for the R°-matrix. For a thorough comparison e.g. the avoided crossings have to be recorded with a better resolution in field strength. Furthermore, special attention has to be paid to reduce field inhomogeneities. -90-

5. Conclusions Stark spectra in barium were studied in excitations of autoionizing Sdnf J=5 Rydberg states using CW laserspectroscopy. In the excitation of these 1=3 states the linear Stark effect became immediately apparent because of their small quantum defects (n=0.074 for n=60). The onset and development of the n=60 manifold was investigated up till field strengths where this manifold merged with the n=61 manifold. In the field higher and lower /-states of opposite parity are dipole-coupled. The states with lower /-values have pronounced quantum defects and strong couplings with their underlying continuum. As a consequence broad resonances in between the manifolds were observed as well as interference patterns in the intensity envelope of the manifolds. These phenomena are typical for autoionizing states and are not present in Stark spectra of bound states. A method for calculating autoionizing spectra in the presence of the field makes use of Multichannel Quantum Defect Theory adapted for the Stark effect. Stark spectra for different Iml components were constructed and compared with experimental observations. It may be concluded that general trends are reproduced but a detailed comparison down to the experimental accuracy at MHz level was not feasible because of insufficient experimental information, especially on the low-/ series. For example narrow resonances observed in between the n=61 and n=60 manifold could be understood as the avoided crossing between the 5d63d level with the outermost states of the n=60 manifold but calculated values for the level separations not reproduced the experimental values. For a stringent test of this MQDT method more accurate experimental data are needed, in particular on the 5dns,5dnp and 5dnd autoionizing series, to deduce correct values for the input parameters required in the calculation.

Also the method of diagonalising the complete energy matrix was applied to calculate energy positions. It turned out that for lml=2 the agreement with results obtained with the MQDT method was excellent, but the diagonalisation procedure could not reproduce accurately the peak positions in m=0 and lml=l spectra. This may be attributed to the fact that the coupling with the continuum is not included in the diagonalisation method. For the spectra investigated especially the Sdns and 5dnp autoionizing series have a pronounced interaction with the continuum, which is only reflected in m=0 and lml=l spectra. Furthermore, with increasing field strength more and more n have to be included in the energy matrix making it too large for any practical purpose. With the proper input parameters the MQDT method can, in principle, calculate accurate Stark spectra for any value of the field. At present a more detailed experimental investigation of the Stark effect in the Sdnf autoionizing series in barium is in progress in our laboratory. -91-

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33. W. Vassen, E.A J.M. Bente and W. Hogervoisu J. Phys. B: At. Mol. Phys. 20 (1987) 2383 34. E.AJ.M. Bente and W. Hogervorst: Phys. Rev. A 36 (1987) 4081 35. C.T.W. Lahaye, T. van der Veldt and W. Hogervorst: submitted for publication 36. E.AJ.M. Bente and C.T.W. Lahaye: unpublished data 37. D. Kteppner, M.G. Liunan and M.L. Zimmerman: In: Rydberg states of atoms and molecules. Cambridge, Cambridge University Press 1983 38. A. Giusb'-Suzor and U. Fano: J. Phys. B: At Mol. Phys. 17 (1984) 215 39. E.AJ.M. Bente and W. Hogervorst: to be published 40. M. Ayman J. Phys. B: At Mol. Phys. 18 (1985) L763 41. W.R.S. Garton and RS. Tomkins: Astrophys. J. 15S (1969) 1219 42. J.P. Coraierade, A.M. Lose and M.A. Baig: J. Phys. B: At Mol. Phys. 18 (1985) 3507 43. C.T.W. Lahaye and W. Hogervorst: submitted to Phys. Rev. A (Chapter 5) -93-

Samenvatting

In dit proefschrift wordt de invloed van statische electrische velden op hoog aangeslagen electrontoestanden van de atomen barium en helium behandeld. Onder invloed van het veld verschuiven en splitsen de atomaire energienivo's. Dit effect wordt meestal geassocieerd met de naam Stark, die het voor het eerst in 1913 waarnam in het waterstofatoom. De hoog aangeslagen zg. Rydbergtoestanden van barium en helium werden bezet vanuit metastabiele toestanden. De excitatie vond plaats d.m.v. absorptie van continu laserlicht. De energie-afstand die met name in helium overbrugd dient te worden voor de excitatie is echter zodanig groot (= 5 eV) dat of met twee-staps excitaties met meerdere lasers, dan wel met frequentie-verdubbeld licht gewerkt moest worden. Ultraviolet-laserlicht werd geproduceerd door m.b.v. een niet-lineair optisch kristal in de trilholte van de kleurstoflaser de frequentie van het zichtbare licht te verdubbelen (of de golflengte te halveren). Om een zo hoog mogelijk scheidend vermogen te krijgen werd de frequentie van het laserlicht m.b.v. externe Fabry-Pérot etalons gestabiliseerd. Om Doppler-effecten zo goed mogelijk te elimineren werden de experimenten uitgevoerd in een opstelling waarbij een gecollimeerde atoombundel loodrecht gesneden wordt door de laserbundel(s). De barium atoombundel werd geproduceerd door verhitting van een oventje waaruit de atomen door een klein gaatje konden ontsnappen. De helium atoombundel werd verkregen door het gas door een capillair te per- sen. Een effectieve bezetting van metastabiele toestanden, zowel voor barium als voor helium, werd verkregen m.b.v. een gasontlading door de bundel. In het interactiegebied was een tweetal condensatorplaten gemonteerd voor het aanleggen van een electrisch veld. Deze waren zodanig opgesteld dat de richting van het electrisch veld loodrecht stond op zowel de atoom- als laserbundel. De Rydbergatomen werden gedetcteerd dm.v. veldionisatie, d.w.z. na excitatie werd het nog maar zwak gebonden electron met een tweede veld van het atoom los- gemaakt. De vrijkomende electronen (barium) of ionen (helium) werden met een electron- multiplier gedecteerd. Deze signalen werden daarna verder verwerkt en geanalyseerd m.b.v. de computer die ook de experiment-besturing regelde.

De onderzochte atomen behoren tot de klasse van de zg. twee-electron systemen. Dit wil zeggen dat er zich twee electronen buiten enkele gevulde electronenschiüen bevinden. Helium is het meest fundamentele twee-electron systeem omdat dit atoom in totaal slechts twee electronen bezit. In barium daarentegen bewegen de twee valentie electronen zich in het veld van een dubbel geladen ion met de xenon edelgasconfiguratie. De meest gebruikelijke, quantum- mechanische aanpak om een veel-deeltjes systeem te beschrijven is de "central field approximation". Hierbij wordt aangenomen dat ieder electron zich onafhankelijk beweegt in het ge- middelde, centrale veld van de positief geladen kern en de andere electronen samen. Ieder individueel electron wordt gekarakteriseerd met een hoofdquantumgetal n; en een -94-

baanimpulsmoment quantumgetal /;. De zg. configuratie wordt dan bepaald door de totale set

van quantumgetallen {n; /;}. Een atoom bevindt zich in een Rydbergtoestand als eén van beide valentie electronen in een hoog aangeslagen toestand (grote n) is gebracht. Een Ryd- bergserie bestaat uit een verzameling energienivo's met vast quantum getal / en toenemend hoofdquantumgetal n. De gebonden Rydbergseries in barium en helium zijn lsn/ en 6sn/ respectievelijk. Bij toenemende n neemt de bindingsenergie van het Rydbergelectron af en uiteindelijk convergeert de serie naar de ionisatielimiet, d,w.z. naar de grondtoestand van het bijbehorende ion. De energie van een toestand wordt, naar analogie met het waterstofatoom, bepaald door (n — |Af)~2 waarbij het quantum defect jij een maat is vour de afwijking met de corresponderende energietoestand in het waterstofatoom. Met toenemende waarde van / neemt de waarde van dit quanturn defect zeer snel af. Er zijn ook Rydbergseries die conver- geren naar aangeslagen toestanden van het ion. In dit geval zijn beide valentie electronen aangeslagen. Laagliggende gebonden toestanden van deze series kunnen het regelmatige gedrag van de gebonden Rydbergseries verstoren. Dit verschijnsel treedt alleen in barium op. Dubbel-aangeslagen toestanden boven de ionisatielimiet worden autoioniserend genoemd omdat ze uiteindelijk in een ion en een electron uiteen zullen vallen. In hoofdstuk 2 worden een aantal methodes behandeld om de wisselwerking tussen atoom en electrisch veld te berekenen. Zolang de invloed van het electrisch veld klein is t.o.v. andere interacties kan het probleem met storingstheorie worden aangepakt. In het algemeen is er voor kleine waarden van / geen eerste orde effect. Alleen het quadratisch Stark-effect, waarbij verschuiving en opsplitsing van energienivo's evenredig is met het kwadraat van het aangelegde veld, wordt dan waargenomen. Met toenemende waarde van / neemt echter de energieafstand tussen toestanden met verschillende /-waarden maar met gelijke n-waarde af. De nivo's zijn dan bijna ontaard en een geleidelijke overgang van quadratisch naar lineair Stark-effect wordt waargenomen met toenemende waarde van /. De (bijna) ontaarding wordt opgeheven door het veld. De Starktoestanden waaieren evenredig met de sterkte van het veld uit en de zg. impulsmoment manifold wordt waargenomen. In barium is bij zwakke velden deze manifold voor / > 3 (zie Hoofdstuk 2) en voor helium reeds voor /äl (zie Hoofdstuk 5) waar te nemen. Een betere manier om manifolds te berekenen is diagonalisatie van de totale energie matrix. Op de diagonaal worden de nul-veld energieën ingevoerd, terwijl de zijdiagonalen de electrische veld bijdrage bevatten. Aangezien het electrisch veld geen (dipool)selectieregel voor n kent heeft dit tot gevolg dat in principe alle toestanden met verschillende n-waarden moeten worden beschouwd. Gelukkig wordt met toenemende energieafstand het bijbehorende matrixelement snel kleiner. De gewenste nauwkeurigheid bepaalt het aantal n-waarden dat uiteindelijk moet worden meegenomen en dus de omvang van de energie matrix. In principe voldoet deze methode uitstekend voor gebonden toestanden. Naarmate echter de veldsterkte -95-

toeneemt zullen meer en meer n-waarden moeten worden meegenomen. De maximale geheu- gen capaciteit van de gebruikte computer zal in de praktijk snel bereikt zijn. Een belangrijke tekortkoming van de diagnalisatie procedure is dat de invloed van het continuum niet in rekening wordt gebracht. Deze invloed is zowel voor autoioniserende toestanden als gebonden toestanden in sterke velden.waar bv. het tunneleffect kan optreden, niet te verwaarlozen. De meest geavanceerde theorie die de interactie van een Rydbergatoom met het electrisch veld beschrijft is de voor het Stark-effect aangepaste Multichannel Quantum Defect Theory (MQDT). Basis idee van deze theorie is de splitsing van de configuratie-ruimte in twee gebieden met ieder hun specifieke gedrag. Te onderscheiden zijn het gebied dat zich beperkt tot de omgeving van de zg. core (kern + gesloten electronschillen), waarin het veel- deeltjes karakter van het systeem tot uiting komt, en het gebied op grote afstand van de core. Hier gedraagt het Rydberg electron zich als een onafhankelijk deeltje bewegend in een Coulomb potentiaal. Met in het laboratorium bereikbare veldsterkten is het niet mogelijk invloed uit te oefenen op de beweging van het electron in de nabijheid van de core. Hier blijft de bij afwezigheid van een veld gebruikelijke beschrijving m.b.v. quantum defecten geldig. Het veld heeft alleen invloed op grote afstand van de core. In dit buitengebied lijkt het Ryd- bergelectron sterk op het waterstofatoom. Het probleem wordt dan ook bepaald door de oplossing van de Schrödingervergelijking voor waterstof in een electrisch veld. MQDT schrijft uiteindelijk voor hoe de oplossingen in de twee gebieden met elkaar verbonden dienen te worden. In hoofdstuk 3 worden de resultaten gepresenteerd van een eerste, verkennende studie van de invloed van het electrisch veld op hoog aangeslagen 6sng (n=40) toestanden in barium met kleine waarden van het quantum defect. De discussie is gewijd aan het ontstaan van de manifolds. Op grond van de kleine, maar essentiële afwijking van waterstof werden allerlei interessante fenomenen verwacht. De excitatie van meerdere IMI componenten (M is de z-component van het totale baanimpulsmoment) tegelijk maakte de spectra dermate complex dat de relevante informatie niet nauwkeurig te verkrijgen was. Eén van die interessante aspecten van waterstofachtig gedrag is het voorkomen van zg. "anti-crossings" wanneer twee toestanden in een veld elkaar naderen. In hoofdstuk 4 worden enkele markante voorbeelden van dergelijke anti-crossings in 3 3 barium behandeld. De anti-crossing van het 6snf F2 nivo met de 6s(n+4)s Si toestand voor M=0 en n = 60 toonde het verwachte gedrag met geleidelijke intensiteits-veranderingen bij nadering van de anti-crossing veldsterkte. Voor de kruising van 6snf 3F2 met 6s(n+4)p ^ bleek echter dat de minimale energie-afstand tussen de twee nivo's kleiner was dan de lijnbreedte van 3F2. De verbreding van dit nivo werd veroorzaakt door kleine inhomogeni- teiten in het veld. Experimenteel resulteerde dit in interferentie-achtige lijnprofielen. Met een aanpassings-procedure kon de feitelijke anti-crossing gereconstrueerd worden uit de -96- waargenomen spectra. Dit leverde de kleinste anti-crossing-afstand op die tot nu toe met laserspectroscopie bepaald kon worden. Om anti-crossings te berekenen met de experimenteel bepaalde MHz nauwkeurigheid moeten voor de diagonalisatieprocedure alle nul-veld energieën van toestanden die een rol spelen zeer precies bekend zijn. Dit leverde voor barium problemen op omdat de uit literatuur beschikbare informatie onvolledig dan wel niet voldoende nauwkeurig is. Een experiment waarin deze moeilijkheden zich niet voordeden wordt beschreven in hoofdstuk 5. Hier worden de experimentele resultaten van Stark effect metingen aan hoge Rydbergtoestanden (n = 40) van helium gepresenteerd. In de discussie wordt nadruk gelegd op anti-crossings tussen Starktoestanden van verschillende manifolds. Vanwege het in fundamenteel opzicht interessante karakter van helium zijn alle nul-veld parameters zeer nauwkeurig bekend. De diagonalisatieprocedure leidde tot goede overeenstemming met de experimentele resultaten. Echter voor het nauwkeurig reproduceren van de anti-crossing tussen Starktoestanden van n=39 en n=41 manifolds bleek dat minstens vijf n-waarden nodig waren in de berekening. Dit betekende het diagonaliseren van een 200 x 200-matrix. In hoofdstuk 6 tenslotte wordt een experiment aan barium besproken waarbij de evolutie van autoioniserende toestanden in een electrisch veld werd gevolgd. Het bijzondere van de onderzochte autoioniserende toestanden is dat hun wisselwerking met het continuum zodanig zwak is dat ze met hoge resolutie continue laserspectroscopie onderzocht kunnen worden. Via de nul-veld excitatie van 5d3/2nf J=5 toestanden (n=60) werden in het veld zowel manifolds (toestanden met />3) als sterk autoioniserende 5dns, 5dnp en Sdnd toestanden aangeslagen. Deze sterk met het continuum gekoppelde toestanden gaven aanleiding tot zowel brede resonanties als smalle interferentiestructuren, met name in het energiegebied tussen twee naburige manifolds. M.b.v. de MQDT methode kon een kwalitatieve verklaring van de waargenomen verschijnselen gegeven worden. Aangezien echter de correcte nul-veld MQDT parameter waarden voor lage /-toestanden (/ < 3) niet nauwkeurig genoeg bekend zijn was een meer kwantitatieve analyse niet mogelijk. Desalniettemin biedt de MQDT versie voor het Stark effect in principe de meest geavanceerde methode om spectra in aanwezigheid van velden te berekenen.

«««««Mc************** Stellingen behorende bij het proefschrift "Stark effect in Rydberg «tates of helium and barium" C.T.W. Lahaye

1. Uit gemeten atomaire niveau-energieën wordt met de Rydberg formule, gebruikmakend van de ionisatie-energie, een quantum defect berekend. De door Freeman en Bjorklund uit experimenteel bepaalde niveau-energieën in strontium afgeleide twee quantum defecten leiden, bij invulling in de Rydberg formule, echter tot tegenstrijdige waarden voor de 4d3/2 ionisatiepotentiaal (60484.50 cm"1 en 60490.28 cm"1). Bovendien wijken deze getallen sterk af van de werkelijke waarde (60481.5 cm"1). R.R. Freeman and G.C. Bjorklund: Phys. Rev. Lett. 40 (1978) 118 CE. Moore: Atomic energy levels vol.2 NSRDS-NBS 35. Washington DC, US Govt. Pr. Off. 1971

2. Om de hyperfijn-geinduceerde n-menging in Rydbergtoestanden te kunnen berekenen wordt door Beigang et al. de totale golffunctie opgesplitst in een deel dat de hyperfijn- structuur bepaalt en een deel dat de overlap van radiële golffuncties tussen twee naburige Rydbergtoestanden beschrijft. Deze beschrijvingswijze is onjuist omdat de menging van fijnstructuur-golffuncties verwaarloosd wordt. R. Beigang, W. Makat, A. Timmermann and PJ. West Phys. Rev. Lett. 51 (1983) 771

3. Coherente XUV (extreem ultra-violet) en VUV ( vacuüm ultra-violet) straling kan geproduceerd worden m.b.v. frequentie-verdrievoudigings-technieken in meer-atomige moleculaire gassen. Dit is in tegenstelling met de algemene opvatting dat alleen atomaire gassen en dampen voor dit doel bruikbaar zijn.

4. De optimale golflengte voor het behandelen van wijnvlekken met milliseconde- laserpulsen ligt niet bij de absorptiepiek van haemoglobine (577 nm) maar bij iets langere golflengten in verband met de dan grotere penetratie-diepte in bloed.

5. Hoog-aangeslagen toestanden van atomen of moleculen kunnen vanuit de grondtoestand worden bezet d.m.v. absorptie van één hoog-energetisch laserfoton dan wel door de gelijktijdige absorptie van een aantal laag-energetische fotonen (multifoton-excitatie). Het verdient echter aanbeveling, daar waar mogelijk, deze toestanden te exciteren m.b.v. één hoog-energetisch foton, verkregen door omzetting van dit aantal laag-energetische fotonen in een niet-lineair optisch proces. 6. Voor spectroscopisch onderzoek aan stoffen onder hoge, statische drukken in een diamantcel is tijd-opgeloste, gestimuleerde Ramanspectroscopie (SRS) te prefereren boven tijd-opgeloste coherente anti-Stokes Ramanspectroscopy (CARS).

7. De door astrofysici veel toegepaste afkorting a.u. voor "astronomical unit" kan bij atoomfysici, die dezelfde afkorting gebruiken voor "atomic unit", leiden tot een verkeerde interpretatie van de afmetingen van ons zonnestelsel. De verwarring wordt nog groter indien men zich realiseert dat dezelfde afkorting soms ook staat voor "arbitrary unit".

8. Medici dienen in hun voordrachten en publicaties af te zien van een verhandeling over de werking van de laser.

5. Een fluorescentie experiment is een vorm van oplichten.

10. De toenemende vervuiling van het milieu leidt tot de vraag: "Wat verdwijnt er het eerst; de kip of het ei?"

11. Zij die beweren dat alleen toegepast wetenschappelijk onderzoek nuttig is dienen zich te realiseren dat zonder fundamenteel wetenschappelijk onderzoek de mens nu in haute couture berevellen gewapend met perfect uitgebalanceerde knotsen zou rondlopen.

C.T.W. Lahaye 18 januari 1989 STARK EFFECT IN RYDBERG STATES OF HELIUM AND BARIUM

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